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ScienceDirectIFAC-PapersOnLine 48-1 (2015) 435–440

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2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.Peer review under responsibility of International Federation of Automatic Control.10.1016/j.ifacol.2015.05.081

Gudrun Mikota et al. / IFAC-PapersOnLine 48-1 (2015) 435–440

© 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Low Frequency Correction of a Multi-degrees-of-freedom Model for Hydraulic Pipeline Systems

Gudrun Mikota*

*Institute of Machine Design and Hydraulic Drives, Johannes Kepler University Linz, 4040 Linz, Austria (Tel: +43-732-2468-6538; e-mail: gudrun.mikota@ jku.at)

Abstract: For hydraulic pipeline systems, a multi-degrees-of-freedom model is developed from the modal decomposition of the transfer function between flow rate excitation and pressure response. Eigenvectors are taken from the undamped case. Natural frequencies and damping ratios are calculated from modal approximations of the individual pipelines with single frequency approximations at the pipeline system resonances in the low frequency range. The multi-degrees-of-freedom pipeline system model is rebuilt from its modal description and evaluated for a network that connects a pump with two closed volumes.

Keywords: Hydraulic pipelines, Approximate analysis, Eigenmode analysis, Eigenvalues, Eigenvectors, Transfer functions

1. INTRODUCTION

Undamped fluid flow in a pipeline is described by the wave equation, which Ingard (1988) treats as a common model for electromagnetic waves on a cable, sound waves in a fluid, longitudinal waves on a solid bar, and torsional waves on a rod. If damping is taken into account, the model for oil hydraulic applications becomes more specific. It can be based on laminar flow conditions since turbulence would increase pressure loss and is usually avoided in oil hydraulics. The transient laminar flow of a compressible Newtonian fluid in a straight circular pipeline was described by transcendental transfer functions from D'Souza et al. (1964) and has further been modelled by rational fraction modal approximations. Such models were published by Almondo et al. (2006), Ayalew et al. (2005), Hsue et al. (1983), Mäkinen et al. (2000), van Schothorst (1997), and Yang et al. (1991); they can be used for time-domain simulation and are well suited to study the dynamic behaviour of individual pipelines. Compared to the transcendental model by D'Souza et al. (1964), Kojima et al. (2002) encountered large errors when they combined modal approximations for the simulation of compound pipeline systems; they therefore suggested to calculate transcendental transfer functions of the entire system and approximate the result in a second step.

For a closed-end pipeline, injected flow rate excitations and resulting pressure responses, Mikota (2013) derived the modal decomposition of the transcendental pipeline model. Transcendental modal transfer functions were approximated by rational fraction expressions, which lead to a multi-degrees-of-freedom description of the pipeline. Mikota (2014) used this model to investigate a specific pipeline network and experienced similar problems as Kojima et al. (2002). However, by comparing transcendental and approximated transfer functions of the network, it became clear that these problems were located in the low frequency

range. They were explained by the fact that for damped pipeline systems, the modal approximation of an individual pipeline is rather inaccurate in the frequency range below the first pipeline resonance. If the pipeline becomes part of a network, this frequency range will contain one or more network resonances, for which the approximation will be wrong.

In this paper, the modal approximations from Mikota (2013) are modified in a way that corrects the low frequency errors for a predefined pipeline system. Proportional damping is enforced on the pipeline system model so that all eigenvectors can be taken from the undamped case. In the low frequency range, natural frequencies and damping ratios are calculated from single frequency approximations of the individual pipelines. Higher natural frequencies and the respective damping ratios are taken from the viscous damping approximaton as used by Mikota (2014). The new method is applied to the pipeline network from Mikota (2014) and leads to a significant improvement of the multi-degrees-of-freedom pipeline system model.

2. EXAMPLE SETUP AND PREVIOUS RESULTS

Fig. 1. Hydraulic pipeline network.

To motivate the necessity of a low frequency correction, example and results from Mikota (2014) are summarized in

8th Vienna International Conference on Mathematical ModellingFebruary 18 - 20, 2015. Vienna University of Technology, Vienna,Austria

Copyright © 2015, IFAC 435


Gudrun Mikota*




1. INTRODUCTION











Gudrun Mikota*




1. INTRODUCTION











Gudrun Mikota*




1. INTRODUCTION











Gudrun Mikota*




1. INTRODUCTION










436 Gudrun Mikota et al. / IFAC-PapersOnLine 48-1 (2015) 435–440

this Section. Figure 1 shows the pipeline network under consideration. It consists of pipeline 1 with l1 = 2.3 m, which is connected to a pump, and pipelines 2 and 3 with l2 = l3 = 3.0 m, each of which leads to a closed volume with Vc = 1 dm3 (e.g. a cylinder volume). The inner radius of all pipelines equals r = 1 cm. The fluid bulk modulus is taken as E = 2⋅109 Pa and the fluid density as ρ = 1000 kg⋅m-3, leading to a speed of sound c = /E 1414 m⋅s-1; the kinematic viscosity is taken as = 5⋅10-5 m2⋅s-1.

Fig. 2. Comparison of transfer functions between flow rate excitation and pressure response at the inlet of pipeline 1. (a): magnitude, (b): phase. line 1: approximated (uncorrected multi-degrees-of-freedom), line 2: transcendental.

The pump injects a defined flow rate excitation at the inlet of pipeline 1. It therefore makes sense to consider the transfer function between flow rate excitation and pressure response. By a comparison of transcendental and approximated transfer functions, Fig. 2 shows how an uncorrected multi-degrees-of-freedom approximation exaggerates the magnitudes at the lower two resonances. A linear amplitude scale is used to demonstrate the extent of the problem.

3. HYDRAULIC PIPELINE SYSTEM MODEL

Compared to Mikota (2014), the hydraulic pipeline system model is rebuilt from a modal description in which some natural frequencies and damping ratios are corrected. Although the underlying multi-degrees-of-freedom model of an individual pipeline features proportional damping, this is not necessarily the case for the assembled multi-degrees-of-freedom model of the hydraulic pipeline system. To keep within the framework of proportional damping, the eigenvectors of the corrected system are assumed to be real and can therefore be taken from the undamped version of the pipeline system model.

3.1 Undamped Case

For an undamped pipeline with closed ends, the modal decomposition of the transfer function between the flow rate

excitation Qex at the axial coordinate xk and the pressure response P at the axial coordinate xj reads

lxn

lxn

sGAlE

AlsE

sxQsxP

k

n

jn

kex

j

coscos)(2

),(),(

1

(1)

with the modal transfer function

,)( 22

lcns

ssGn

(2)

where l denotes the length of the pipeline, A is the pipeline cross-sectional area, E is the bulk modulus of the fluid, and c is the speed of sound.

The mobility function of a mechanical system is defined as the frequency response function between excitation force and velocity response. For undamped and proportionally damped systems, Ewins (2000) derives the description of the mobility function in terms of eigenvalues and mass-normalized eigenvectors.

In the following, flow rate excitation and pressure are considered at the discrete coordinates x1, x2, …, xN+1. If (1) is truncated after mode N, the respective frequency response function can be recognized as mobility function of an undamped mechanical multi-degrees-of-freedom system with mass-normalized (N+1)×1 eigenvectors

TAlE 1110 (3)

and

AlE

n2

,,,2,1

,coscoscos 121

Nnlxn

lxn

lxn

TN

(4)

and natural frequencies

.,,1,0, Nnlcn

n

(5)

Using the (N+1)×(N+1) matrix

N 10 (6)

and the (N+1)×(N+1) diagonal matrix ][ 2nω , if follows from

the orthogonality relations for mass-normalized eigenvectors that the mass matrix of the equivalent mechanical model reads

,11 TM (7)

and the stiffness matrix becomes

,][ 121 n

Tu ωK (8)

MATHMOD 2015February 18 - 20, 2015. Vienna, Austria

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Gudrun Mikota et al. / IFAC-PapersOnLine 48-1 (2015) 435–440 437

this Section. Figure 1 shows the pipeline network under consideration. It consists of pipeline 1 with l1 = 2.3 m, which is connected to a pump, and pipelines 2 and 3 with l2 = l3 = 3.0 m, each of which leads to a closed volume with Vc = 1 dm3 (e.g. a cylinder volume). The inner radius of all pipelines equals r = 1 cm. The fluid bulk modulus is taken as E = 2⋅109 Pa and the fluid density as ρ = 1000 kg⋅m-3, leading to a speed of sound c = /E 1414 m⋅s-1; the kinematic viscosity is taken as = 5⋅10-5 m2⋅s-1.

Fig. 2. Comparison of transfer functions between flow rate excitation and pressure response at the inlet of pipeline 1. (a): magnitude, (b): phase. line 1: approximated (uncorrected multi-degrees-of-freedom), line 2: transcendental.

The pump injects a defined flow rate excitation at the inlet of pipeline 1. It therefore makes sense to consider the transfer function between flow rate excitation and pressure response. By a comparison of transcendental and approximated transfer functions, Fig. 2 shows how an uncorrected multi-degrees-of-freedom approximation exaggerates the magnitudes at the lower two resonances. A linear amplitude scale is used to demonstrate the extent of the problem.

3. HYDRAULIC PIPELINE SYSTEM MODEL

Compared to Mikota (2014), the hydraulic pipeline system model is rebuilt from a modal description in which some natural frequencies and damping ratios are corrected. Although the underlying multi-degrees-of-freedom model of an individual pipeline features proportional damping, this is not necessarily the case for the assembled multi-degrees-of-freedom model of the hydraulic pipeline system. To keep within the framework of proportional damping, the eigenvectors of the corrected system are assumed to be real and can therefore be taken from the undamped version of the pipeline system model.

3.1 Undamped Case

For an undamped pipeline with closed ends, the modal decomposition of the transfer function between the flow rate

excitation Qex at the axial coordinate xk and the pressure response P at the axial coordinate xj reads

lxn

lxn

sGAlE

AlsE

sxQsxP

k

n

jn

kex

j

coscos)(2

),(),(

1

(1)

with the modal transfer function

,)( 22

lcns

ssGn

(2)

where l denotes the length of the pipeline, A is the pipeline cross-sectional area, E is the bulk modulus of the fluid, and c is the speed of sound.

The mobility function of a mechanical system is defined as the frequency response function between excitation force and velocity response. For undamped and proportionally damped systems, Ewins (2000) derives the description of the mobility function in terms of eigenvalues and mass-normalized eigenvectors.

In the following, flow rate excitation and pressure are considered at the discrete coordinates x1, x2, …, xN+1. If (1) is truncated after mode N, the respective frequency response function can be recognized as mobility function of an undamped mechanical multi-degrees-of-freedom system with mass-normalized (N+1)×1 eigenvectors

TAlE 1110 (3)

and

AlE

n2

,,,2,1

,coscoscos 121

Nnlxn

lxn

lxn

TN

(4)

and natural frequencies

.,,1,0, Nnlcn

n

(5)

Using the (N+1)×(N+1) matrix

N 10 (6)

and the (N+1)×(N+1) diagonal matrix ][ 2nω , if follows from

the orthogonality relations for mass-normalized eigenvectors that the mass matrix of the equivalent mechanical model reads

,11 TM (7)

and the stiffness matrix becomes

,][ 121 n

Tu ωK (8)


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see Ewins (2000).

The assumption of closed ends corresponds to free boundaries in the equivalent mechanical system and still allows the injection of flow rate excitations, which correspond to mechanical excitation forces.

For a system consisting of several pipelines, the local mass and stiffness matrices )(i

lM and )(iulK of the individual

pipelines are enlarged to obtain the respective global matrices )(i

gM and )(iugK according to Ginsberg (2001). These can be

assembled into the global mass matrix

i

igg

)(MM (9)

and the global stiffness matrix

i

iugug

)(KK (10)

of the undamped pipeline system. Ewins (2000) and Ginsberg (2001) give the respective equation of motion for mechanical systems.

The eigenvalue problem

0θKM ugg2 (11)

yields the real eigenvectors uθ and the undamped natural frequencies u .

3.2 Viscous Damping Approximation

For a closed-end pipeline with laminar flow, the modal decomposition of the transcendental transfer function between flow rate excitation and pressure response still follows (1), but the modal transfer function assumes the transcendental form

22

2

)(1

)(

sflcns

ssGn

(12)

with

,)(

2

0

sirJ

sirJsf (13)

where 0J and 2J are Bessel functions of first kind, denotes the kinematic viscosity, and r is the pipeline radius.

Mikota (2013) obtained a viscously damped approximation of the transcendental modal transfer function by arguing that for lightly damped pipelines, its major contribution to the modal series (1) appears in the vicinity of the undamped resonance (5). In )(sGn , s and ni were exchanged twice to arrive at the approximated modal transfer function

,2

)( 200

2nnn

nv ssssG

(14)

where

20 )(

1Ren

nn if (15)

and

.

)(1Re2

)(1Im

2

2

n

nn

if

if

(16)

If )(sGn is replaced by )(sGnv , the frequency domain form of the truncated modal series (1) becomes the mobility function of a proportionally damped mechanical multi-degrees-of-freedom system with mass-normalized eigenvectors (3) and (4), undamped modal natural frequencies 000 and (15) for n=1,2,…,N, and modal damping ratios 00 and (16) for n=1,2,…,N.

The mass matrix of the equivalent mechanical model is given by (7) using (6) as in the undamped case; with the (N+1)×(N+1) diagonal matrices ][ 2

0n and ]2[ 0nn , the stiffness matrix becomes

120

1][

nT

v ωK (17)

and the damping matrix reads

.]2[ 10

1 nn

Tv ωC (18)

For a hydraulic pipeline system, one can proceed as described by Mikota (2014). The local mass, stiffness, and damping matrices )(i

lM , )(ivlK , and )(i

vlC of the individual pipelines are

enlarged to obtain the respective global matrices )(igM , )(i

vgK ,

and )(ivgC . These are assembled into the global mass

matrix (9), the global stiffness matrix

i

ivgvg

)(KK (19)

and the global damping matrix

i

ivgvg

)(CC (20)

of the hydraulic pipeline system.

The first-order eigenvalue problem

vv BA 0θθ

(21)

with the matrices

0MMC

Ag

gvgv (22)

and


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g

vgv M0

0KB (23)

results in complex conjugate eigenvalues v and eigenvec-tors vθ . For undercritical damping, the eigenvalues v are related to the undamped natural frequencies v and the damping ratios v by

vvv Re (24)

and

.1Im 2vvv (25)

3.3 Single Frequency Approximation

Instead of fitting the transcendental modal transfer function )(1 sG in the vicinity of the first undamped resonance 1 , one

may choose a different angular frequency a for which s and

ai are exchanged. This results in the approximation

201011

21 2)(

aaaa ss

ssG

(26)

with

2101 )(

1Rea

a if (27)

and

.

)(1Re2

)(1Im

2

21

1

aa

aa

if

if

(28)

Such an approximation may be rather inaccurate in the vicinity of the first undamped resonance 1 ; a viscous damping approximation with )(1 sG a instead of )(1 sG v will in general lead to poor results for an individual pipeline. However, for 1 a , this approximation is accurate in the vicinity of a since higher modes do not contribute much in the frequency range below the first pipeline resonance. The frequency domain form of the truncated modal series (1) then becomes the mobility function of a proportionally damped mechanical multi-degrees-of-freedom system with mass-normalized eigenvectors (3) and (4), undamped modal natural frequencies ,000 (27), and (15) for n=2,3,…N, and modal damping ratios ,00 (28), and (16) for n=2,3,…N.

For a system of several pipelines, one may wish a precise approximation at a specific angular frequency a . The approximation of an individual pipeline then depends on the relation between a and the first undamped resonance 1 . If

1 a , the local stiffness and damping matrices )(ialK and

)(ialC of the pipeline are obtained as described in

Subsection 3.2 so that )()( ivl

ial KK and )()( i

vli

al CC . If

1 a , the local stiffness matrix )(ialK is calculated from

(17) by replacing a01 for the first undamped modal natural

frequency 01 , and the local damping matrix )(ialC is

determined from (18) by further replacing a1 for the first

modal damping ratio 1 . The local mass matrix )(ilM is

always given by (7).

The determination of eigenvalues follows the procedure from Subsection 3.2. Local mass, stiffness, and damping matrices of the individual pipelines are enlarged to obtain the respective global matrices and assembled into the global mass, stiffness, and damping matrices gM , agK , and agC of the pipeline system. The corresponding eigenvalue problem yields the complex conjugate eigenvalues a . If a is a pipeline system resonance frequency, the single frequency approximation can be expected to provide accurate eigenvalues c near ai .

4. LOW FREQUENCY CORRECTION

The viscous damping approximation of the hydraulic pipeline system can in principle be used above the highest first resonance of the individual pipelines. Of course, there is an upper frequency limit that can be extended by including more pipeline modes, but the focus of this paper is a suitable description in the low frequency range. In particular, the behaviour at the pipeline system resonances should be captured. Inaccuracies in the modal approximations of the individual pipelines must be avoided around these specific frequencies. A single frequency approximation is therefore used for each pipeline system resonance below the highest first resonance of the individual pipelines; its contribution to the corrected pipeline system model consists of two complex conjugate eigenvalues c , from which an undamped natural frequency c and a damping ratio c can be extracted with

ccc Re (29)

and

.1Im 2ccc (30)

Above the highest first resonance of the individual pipelines, undamped natural frequencies and damping ratios are transferred from the viscous damping approximation. In this way, all eigenvalues of the corrected pipeline system model are collected.

Although both viscous damping and single frequency approximation of an individual pipeline feature proportional damping, this is not necessarily the case with the respective global models of the pipeline system. The latter may exhibit complex eigenvectors, which do not fit into the framework of (7), (17), and (18). However, Ewins (2000) states that significant complexity in a structure's modes will only arise if two or more modes are "close". For the hydraulic pipeline system, it is assumed that all modes are sufficiently separated, and proportional damping is enforced on the


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Gudrun Mikota et al. / IFAC-PapersOnLine 48-1 (2015) 435–440 439

g

vgv M0

0KB (23)

results in complex conjugate eigenvalues v and eigenvec-tors vθ . For undercritical damping, the eigenvalues v are related to the undamped natural frequencies v and the damping ratios v by

vvv Re (24)

and

.1Im 2vvv (25)

3.3 Single Frequency Approximation

Instead of fitting the transcendental modal transfer function )(1 sG in the vicinity of the first undamped resonance 1 , one

may choose a different angular frequency a for which s and

ai are exchanged. This results in the approximation

201011

21 2)(

aaaa ss

ssG

(26)

with

2101 )(

1Rea

a if (27)

and

.

)(1Re2

)(1Im

2

21

1

aa

aa

if

if

(28)

Such an approximation may be rather inaccurate in the vicinity of the first undamped resonance 1 ; a viscous damping approximation with )(1 sG a instead of )(1 sG v will in general lead to poor results for an individual pipeline. However, for 1 a , this approximation is accurate in the vicinity of a since higher modes do not contribute much in the frequency range below the first pipeline resonance. The frequency domain form of the truncated modal series (1) then becomes the mobility function of a proportionally damped mechanical multi-degrees-of-freedom system with mass-normalized eigenvectors (3) and (4), undamped modal natural frequencies ,000 (27), and (15) for n=2,3,…N, and modal damping ratios ,00 (28), and (16) for n=2,3,…N.

For a system of several pipelines, one may wish a precise approximation at a specific angular frequency a . The approximation of an individual pipeline then depends on the relation between a and the first undamped resonance 1 . If

1 a , the local stiffness and damping matrices )(ialK and

)(ialC of the pipeline are obtained as described in

Subsection 3.2 so that )()( ivl

ial KK and )()( i

vli

al CC . If

1 a , the local stiffness matrix )(ialK is calculated from

(17) by replacing a01 for the first undamped modal natural

frequency 01 , and the local damping matrix )(ialC is

determined from (18) by further replacing a1 for the first

modal damping ratio 1 . The local mass matrix )(ilM is

always given by (7).

The determination of eigenvalues follows the procedure from Subsection 3.2. Local mass, stiffness, and damping matrices of the individual pipelines are enlarged to obtain the respective global matrices and assembled into the global mass, stiffness, and damping matrices gM , agK , and agC of the pipeline system. The corresponding eigenvalue problem yields the complex conjugate eigenvalues a . If a is a pipeline system resonance frequency, the single frequency approximation can be expected to provide accurate eigenvalues c near ai .

4. LOW FREQUENCY CORRECTION

The viscous damping approximation of the hydraulic pipeline system can in principle be used above the highest first resonance of the individual pipelines. Of course, there is an upper frequency limit that can be extended by including more pipeline modes, but the focus of this paper is a suitable description in the low frequency range. In particular, the behaviour at the pipeline system resonances should be captured. Inaccuracies in the modal approximations of the individual pipelines must be avoided around these specific frequencies. A single frequency approximation is therefore used for each pipeline system resonance below the highest first resonance of the individual pipelines; its contribution to the corrected pipeline system model consists of two complex conjugate eigenvalues c , from which an undamped natural frequency c and a damping ratio c can be extracted with

ccc Re (29)

and

.1Im 2ccc (30)

Above the highest first resonance of the individual pipelines, undamped natural frequencies and damping ratios are transferred from the viscous damping approximation. In this way, all eigenvalues of the corrected pipeline system model are collected.

Although both viscous damping and single frequency approximation of an individual pipeline feature proportional damping, this is not necessarily the case with the respective global models of the pipeline system. The latter may exhibit complex eigenvectors, which do not fit into the framework of (7), (17), and (18). However, Ewins (2000) states that significant complexity in a structure's modes will only arise if two or more modes are "close". For the hydraulic pipeline system, it is assumed that all modes are sufficiently separated, and proportional damping is enforced on the


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corrected global pipeline system model. Taking the eigenvectors uθ from the undamped case, the pipeline system model can then be rebuilt in the sense of (7), (17), and (18). To begin with, the mass-normalized eigenvectors unθ are required. The matrix unθ composed of mass-normalized eigenvectors and the global mass matrix gM must satisfy the orthogonality relation behind (7); it follows that

,1

ugT

uuun θMθθθ (31)

where uθ is a matrix composed of arbitrarily scaled eigenvectors of the undamped pipeline system and the square root is taken per matrix element. Squared natural frequencies

2c and 2

v are collected into a diagonal matrix in the order of the corresponding eigenvectors; the values cc2 and vv2 form a diagonal matrix D in the same order. The global mass matrix of the hydraulic pipeline system is still given by (9); the corrected global stiffness matrix becomes

11 un

Tuncg θθK (32)

and the corrected global damping matrix reads

.11 un

Tuncg θDθC (33)

If the flow rates injected at all system nodes are collected into a global flow rate excitation vector gex,Q and the pressures at all system nodes are collected into a global pressure vector

gP , the low frequency correction of the multi-degrees-of-freedom model for hydraulic pipeline systems is described by

.,2

gexgcgcgg sss QPKCM (34)

Consequently, the global transfer function matrix between flow rate excitation and pressure response can be calculated as

.12

cgcgg sss KCM (35)

5. NUMERICAL EVALUATION

For the hydraulic pipeline network described in Section 2, the first resonance of pipeline 1 lies at 307.4 Hz, and the first resonance of pipelines 2 and 3 lies at 235.7 Hz. Numerical investigations are performed with mesh nodes at every 0.1 m. An undamped eigenvalue analysis of the network yields natural frequencies at 63.5 Hz, 106.3 Hz, and 202.2 Hz, which are below the first resonances of pipelines 1, 2, and 3, and a natural frequency at 259.0 Hz, which is still below the first resonance of pipeline 1. Single frequency approxi-mations are used at these frequencies, and the viscous damping approximation above. From 500 Hz upwards, 10 % damping is assumed to capture the contribution of high frequency modes while avoiding the tedious transfer of individual modal parameters.

Fig. 3. Comparison of transfer functions between flow rate excitation and pressure response at the inlet of pipeline 1. (a): magnitude, (b): phase. line 1: approximated (low frequency correction), line 2: transcendental.

In Fig. 3, the approximation resulting from such a low frequency correction is compared to the transcendental transfer function between flow rate excitation and pressure response at the inlet of pipeline 1. Although the match is not yet perfect, a considerable improvement has been achieved against the uncorrected multi-degrees-of-freedom model used in Fig. 2. In particular, the magnitudes at the lower two resonances fit much better, which is a consequence of correcting the damping ratios.

The transfer functions plotted in Fig. 3 obviously contain resonances near 106.3 Hz and 202.2 Hz; however, the other two low frequency network resonances near 63.5 Hz and 259.0 Hz do not appear. This must be due to the fact that the corresponding mode shapes exhibit a node at the inlet of pipeline 1 and can thus not be excited by a flow rate excitation injected at this point. It can be seen from Fig. 1 that the pipeline network is symmetric, which gives rise to a number of modes where pipeline 1 is a nodal line. It is further clear that all boundaries are free, resulting in an eigenvalue that is zero and together with (35) explains the fact why the phase starts at -90°. The phase lifts between the resonances stem from the denominator of the factorized transfer function and are characteristic of a transfer function between two quantities in the same position, see Ewins (2000).

6. CONCLUSIONS

By using single frequency approximations at the lower system resonances, a fairly accurate multi-degrees-of-freedom model for hydraulic pipeline systems has been established. This model is described by a system of linear second order differential equations with constant coefficients and is therefore suited for time-domain simulation. Apart from that, it gives insight into the modal properties of the pipeline system, which helps understanding the dynamic system behaviour.

In practical applications, hydraulic pipeline systems are combined with valves, cylinders, accumulators, and other


439


components. Although the example given in this paper only mentions two closed volumes, it is always possible to add dynamic relationships between pressure and flow rate as boundary conditions of the pipeline network; they can be linked with the global pressure vector and the global flow rate excitation vector and thus describe the effects of various hydraulic components coupled to the network.

The modal analysis of hydraulic pipelines was originally motivated by the demand for a theory on which the experimental modal analysis of hydraulic pipelines could be based. With the low frequency correction, the accuracy of damped pipeline system models is significantly increased. The low frequency correction thus consolidates the theoretical basis for the experimental modal analysis of hydraulic pipeline systems.

Although in the low frequency range, the essential step has been done, the multi-degrees-of-freedom model for hydraulic pipeline systems could still be developed further. The approximation would probably be improved by accepting complex eigenvectors and dropping the assumption of proportional damping. For higher frequencies, it would be worth investigating the effects of discretisation and coupling modal approximations.

REFERENCES

Almondo, A. and Sorli, M. (2006). Time domain fluid transmission line modelling using a passivity preserving rational approximation of the frequency dependent transfer matrix. International Journal of Fluid Power, Vol. 7, No. 1, pp. 41-50.

Ayalew, B. and Kulakowski, B.T. (2005). Modal approximation of distributed dynamics for a hydraulic transmission line with pressure input - flow rate output causality. Transactions of the ASME - Journal of Dynamic Systems, Measurement, and Control, Vol. 127, pp. 503-507.

D'Souza, A.F. and Oldenburger, R. (1964). Dynamic response of fluid lines. Transactions of the ASME - Journal of Basic Engineering, Vol. 86, pp. 589-598.

Ewins, D.J. (2000). Modal Testing: Theory, Practice, and Application. Second Edition, Research Studies Press Ltd., Baldock, Hertfordshire, England.

Ginsberg, J.H. (2001). Mechanical and Structural Vibrations. John Wiley & Sons, Inc.

Hsue, C.Y. and Hullender, D.A. (1983). Modal approximations for the fluid dynamics of hydraulic and pneumatic transmission lines. In Fluid Transmission Line Dynamics, pp. 51-77. Special Publication for the ASME Winter Annual Meeting, Boston, Massachusetts.

Ingard, K.U. (1988). Fundamentals of waves and oscillations. Cambridge University Press.

Kojima, E., Shinada, M., and Yu, J. (2002). Development of accurate and practical simulation technique based on the modal approximations for fluid transients in compound fluid-line systems. International Journal of Fluid Power, Vol. 3, No. 2, pp. 5-15.

Mäkinen, J., Piché, R., and Ellman, A. (2000). Fluid transmission line model using a variational method.

Transactions of the ASME - Journal of Dynamic Systems, Measurement, and Control, Vol. 122, pp. 153-162.

Mikota, G. (2013). Modal analysis of hydraulic pipelines. Journal of Sound and Vibration, Vol. 332, pp. 2794-3805.

Mikota, G. (2014). A multi-degrees-of-freedom model for hydraulic pipeline systems. In The 9th International Fluid Power Conference, Aachen, Germany.

van Schothorst, G. (1997). Modelling of long-stroke hydraulic servo-systems for flight simulator motion control and system design. PhD Thesis, Delft University of Technology, The Netherlands.

Yang, W.C. and Tobler, W.E. (1991). Dissipative modal approximation of fluid transmission lines using linear friction model. Transactions of the ASME - Journal of Dynamic Systems, Measurement, and Control, Vol. 113, pp. 152-162.


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