An insight into the dynamics of a rigid rotor on two-lobe wave squeeze film damper

Accepted Manuscript An insight into the dynamics of a rigid rotor on two-lobe wave squeeze film damper Giovanni Adiletta PII:

S0301-679X(17)30317-1

DOI:

10.1016/j.triboint.2017.06.029

Reference:

JTRI 4789

To appear in:

Tribology International

Received Date: 23 January 2017 Revised Date:

24 April 2017

Accepted Date: 19 June 2017

Please cite this article as: Adiletta G, An insight into the dynamics of a rigid rotor on two-lobe wave squeeze film damper, Tribology International (2017), doi: 10.1016/j.triboint.2017.06.029. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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An insight into the dynamics of a rigid rotor on two-lobe wave squeeze film damper Giovanni AdilettaI

Abstract

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Differently from journal bearings and to author’s knowledge, the adoption of lobe geometry in the bearing bore of squeeze film dampers (SFD) has not received any research interest, except for some

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recent study due to the present author. In this study, a relatively simple system, represented by a rigid, unbalanced rotor on two-lobe SFD, is theoretically investigated by means of numerical continuation. Despite being restricted to periodic solutions, this numerical procedure permits an efficient characterization of the system’s bifurcating behaviour, suitably taking into account the shape parameters of the wave profile given to the bore of lobed damper bearing. In particular, the influence of the bearing angular position on the synchronous (1-S) and order-two subharmonic solutions (1/2-S) has been analyzed, assuming the rotor speed as the bifurcation parameter, under

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different conditions of bearing operation that are expressed by means of the bearing parameter f. The results are compared to the corresponding behaviour of the rotor on dampers with circular

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bearing (CB). The bifurcation structure shows to be rather influenced by both the angular position and f. With regard to the 1-S, the two-lobe geometry emphasizes some effects obtained raising f in

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the presence of CB. Some other effects show to be distinctive features of the lobed geometry, like the disappearance of the torus bifurcation and the length modification (as increase or decrease) of the segments representing unstable behaviour. A significant simplification in the bifurcation paths of the 1/2-S, as compared to the CB set-up, is achieved at higher f values, especially through suitable

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Dipartimento di Ingegneria Industriale, Universita’ degli Studi di Napoli "Federico II", Napoli, Italy

angular positions. Contrast with this outcome the relatively involved path-patterns obtained for the sub-harmonic responses at the lowest f value. The study represents an insight with respect to the author’s previous works on this subject, which were mainly focused on the characterization of the synchronous behaviour.

Keywords: lubrication, squeeze film damper, wave bearings, bifurcation analysis, numerical continuation

Preprint submitted to Tribology International

June 20, 2017

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Nomenclature

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= y¯s /C coord. of the journal center, m vertical coord. without weight, m = y¯0 +f s gravity residual, m angular coordinate within bearing = L/D, bearing aspect ratio dynamic viscosity, Pa·s rotor static unbalance, m viscous damping coefficient, N·s·m−1 = ωt dimensionless time angular phase in Eq. (2), rad angular speed, rad s−1 3 /(mC 3 ) frequency, rad·s−1 = µRL p = pg/C frequency, rad·s−1 = k/m natural frequency, rad·s−1 = ω/ω R speed parameter

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u2,s x ¯, y¯ y¯0 y¯s δ λ µ ρ σ τ ϕ ω ωB ωg ωR Ω

B amplitude in Eq. (2) C radial clearance, m D = 2R bearing diameter, m f = ω B /ω R bearing parameter fs = −m g/k: static deflection, m f SFx ,f SFy =F SFx /F ∗ , F SFy /F ∗ F SFx ,F SFy film forces, N F∗ = pRL reference pressure force, N g gravity acceleration, m·s−2 h film thickness, m k stiffness of the damper springs, N·m−1 L axial length of the bearing, m m half-mass √ of the rotor, kg q = σ/(2 mk) viscous damping factor u1 ,u2 = x ¯/C, y¯/C u3 ,u4 dimensionless derivatives of u1 and u2

1. Introduction

The way a lubricating oil works within the bearing units of rotating machinery, putting into play its hydrodynamic performances, may be roughly classified in two categories. In the

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first one, the lubricant mainly exerts a load supporting action, thereby enabling the basic

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feature of the bearing support wherein it is employed. In the second category, the fluid film essentially provides a damping action, as in the case of squeeze film dampers. Both ways can

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be jointly adopted within the same support. A common example in this regard is represented by semi-floating ring bearings for automotive turbochargers [1] .

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Whatever the way, the nonlinear film forces are peculiar as far as the dynamic behaviour of the supported rotor is concerned. A number of quantities and factors turn out to influence the hydrodynamics behind the said forces. The clearance space, where the fluid film

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operates, plays of course a fundamental role, with a shape that may be constant or, more generally, variable, depending on geometry and relative motion of its boundary surfaces. The

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investigation particularly addressed to these geometrical aspects and their fluid-dynamic effects has led, in the course of time, to the development of several, distinct types of bearings, thereby enhancing the basic device consisting of an annular clearance, bounded by the two near-concentric, cylindrical surfaces of the shaft-bearing pair. The common fixed, multi-pad journal bearings [2–6] and tilting-pad journal bearings [7, 8] are well known examples that

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employ supporting oil films whose performances benefit, especially in terms of stability, from I

Correspondence to: Dipartimento di Ingegneria Industriale, Via Claudio,21 - 80125, Napoli, Italy. Email address: [email protected] (Giovanni AdilettaI )

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the modification of the basic bearing clearance, through a segmentation achieved by means of fixed or pivoted, tilting pads. In the latter bearing type, however, the pad mobility greatly reduces the destabilizing forces, thereby assuring high dynamic performances of the supported 30

high-speed turbomachinery, at the cost of making the device more involved and expensive than its fixed-pad counterpart. Furthermore, the dependence of the bearing action on a multiplicity

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of layout parameters, which requires a careful design of tilting-pad journal bearings, makes the onset of nonsynchronous responses, including chaos, anything but unlikely. By contrast, the relative simplicity and the consequent low cost have represented a key factor in supporting 35

fixed-pad solutions and nurturing the related research, aimed at improving the quality of the dynamic responses, especially as regards the stability. It is worth remarking that these aspects

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have further encouraged a development of non-circular geometries for gas bearings [9]. A typical feature shared by common multi-pad (or, equivalently, multi-lobe) profiles is

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represented by the offset with respect to the reference concentric pad positions, which can be achieved in a relatively straightforward manner. The deriving preload effect and its effectiveness as a mean to counteract the whirl/whip onset have been the subject of a number of investigations. Very frequently, the related stability analyses have been carried out by determining the instability threshold, with use of linearized dynamic coefficients. Akkök and Ettles [10, 11], for instance, studied how the oil whirl was influenced by circular, elliptical or offset-halves profiles of the bearing, assessing, both theoretically and experimentally, the beneficial role of preloading.

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Journal bearings with non-circular geometries have been theoretically and experimentally studied over the years while deepening several related issues. Non-laminar flow conditions, the

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thermal regime, the influence of pressurization and inlet/discharge system (with distinctive aspects represented by presence, position and dimensions of holes, grooves etc.), the adoption of steps, dams and pockets, the recourse to micropolar fluids, the use of compliant, elastic

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or viscoelastic surfaces, are some regarding examples. Hashimoto et al. [12] theoretically and experimentally studied the static and dynamic characteristics of elliptical bearings in turbulent flow, observing the stability increase as an effect of higher ellipticity ratios. A 55

model for calculating oil film forces within fixed, multi-pad journal bearings, based on the free boundary theory and the variational method, was presented in [13], assessing the time-saving efficiency and accuracy of this approach by comparison with the finite element approach. This same model was adopted in [14] to carry out a nonlinear instability analysis, providing a characterization of the nonlinear behaviour by means of bifurcation diagrams, orbits and

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Poincaré sections obtained through brute force integration of the system equations. Mehta et al. [15] studied a 4-LB with pressure dams, Rao et al. [16] examined the performance

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characteristics of a tri-taper bearing, while Feng and Hahn [17] compared the static and dynamic characteristics obtained with two types of elliptical, pocket bearings. Compliant elliptical bearings were theoretically investigated by Jain et al. [18], who assumed babbit 65

and bronze as the liner materials. They found that deformation increased the load capacity while enhancing stability at lower eccentricities. Compliant polymeric (PTFE) liners were

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investigated by Kuznetsov and Glavatskih [19] adopting theoretical models with perturbation of flow pressure and liner deformation. The presence of an oil chamber placed beneath a flexible pad built in the bearing wall, thereby allowing an active, pressure-controlled operation, 70

characterized the flexible sleeve bearing that was examined in [20], both theoretically and

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experimentally.

The stress on geometrical aspects, given by the research, as a key factor to improve the performance of liquid-lubricated journal bearings, can be further recognized in the published

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investigations carried out about bearing models (either practical or theoretical) that adopt textured surfaces, continuous non-circular geometries, variable geometries and shape optimization. Surface texturing was claimed in [21] as a potential major component in future bearing structure design. Continuous, non-circular profiles, consisting in multi-lobe, waving shapes of the bearing bore, were originally proposed for gas bearings and have been extensively studied also in the liquid lubricated versions [22]. A further, continuous, multi-lobe geometry is represented by the perycicloidal profile [23]. A patented device consisting in a journal bearing

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with variable geometry, capable of suppressing rotor vibration during the passage through

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resonance, is the subject of the investigation reported in [24]. The importance ascribed to the clearance geometry is also the pivotal aspect shared by the

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studies aimed at determining optimized shapes for the bearing bore. Pang et al. [25] made use of a genetic algorithm to determine optimized bearing surfaces, which turned eventually out to be wavy in both circumferential and axial directions. Investigations about the dynamic

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consequences of wear on bearing surfaces [26], are also worthy of mention, concerning the importance assumed by the geometry of the clearance space. When the lubricant liquid is mainly adopted as the damping mean within squeeze film

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dampers, the flow characteristics change with respect to the journal bearing case, but are still conditioned by the clearance geometry and the particular shape of the bearing surfaces, with consequent influences onto the dynamic response of the supported rotor. In this regard, it is worth to remark the nonlinearity affecting the damper response and the behavior of the whole rotor-support system, which is to be ascribed to the forces exerted by the lubricant. Nonlinear

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effects, like multi-stable operation and instabilities leading to nonsynchronous whirl or jumps within the damper bearings, are well known and studied [27, 28]. These effects turn out to be

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particularly influenced by those system parameters that eventually condition the lubricant flow and its forces. The role of rotor unbalance, bearing clearance and aspect ratio, oil film feeding and discharge, cavitation, inertia and turbulence, system mass and stiffness distribution, etc., 100

about the system dynamics, has been pointed out and thoroughly investigated in literature (for instance [29, 30]). In particular, nonlinear analysis carried out through numerical continuation

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demonstrated to be rather effective to clear the dependence of bifurcation changes in the dynamic response of the rotor-support system on different parameters [31, 32]. However, notwithstanding this context, the investigation about the effects of particular geometries and 105

shapes, for the damper clearance and surfaces, turns out to be relatively poor, in contrast to the quantity and variety of analyses and devices mentioned in the above review focused

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on journal bearings. Besides a number of studies regarding the geometry of oil-supply and discharge elements (i.e. holes and grooves), relatively few investigations have been addressed

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to unconventional geometries of the damper gap. The dual clearance [33–35], the multisqueeze [36], and the integral [37, 38] squeeze film dampers are outstanding, related examples. The patented device in [39] presented a clearance that was defined by at least two nonconcentric cylindrical wall portions, thereby giving the chamber a multi-lobe configuration. A further, patented damper is worth mentioning [40]. In this device, the outer surface of a bearing cartridge, representing the internal wall of the damper gap, presents some chambers and tapered zones, suitably placed and aimed at modifying locally the damping forces.

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In the present paper, a continuous, two-lobe, wave profile of the bearing bore is assumed

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in the model of SFD (hereinafter referred to as 2LB) under theoretical investigation. This damper, equipped with retaining springs, supports an unbalanced rigid rotor affected by gravity residual. The unsymmetrical shape of the bore makes its angular position influence the damper response. A further bearing parameter, characteristic of the adopted wave profile,

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is represented by the wave amplitude. A bifurcation analysis of the nonlinear dynamic be-

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haviour, under different sets of system parameters, is carried out by means of the numerical continuation, adopting the rotor speed as the main bifurcation parameter. Previous work on the subject was reported in [41, 42]. In [41] the main effects of the lobe

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geometry on the synchronous solution (1-S) and its stability were shown. The results obtained in [42] showed that, with respect to the circular bearing, an increase in the wave amplitude, for a given angular position, enhanced the stability about resonance. On the contrary, an increase in the length of the speed interval interested by instability of the 1-S was observed in the post-resonance zone, where the 2LB favoured a conversion of quasi-periodic into 1/2-S

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behaviour. These effects turned out to be modulated by the angular position. The presence of 1/3-subharmonic and chaotic behaviour was also shown in [42], even though without a

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systematic analysis about the parametric dependence. The role of the gravity residual was also observed in that paper, assuming in turn mild, moderate, and severe conditions. An effort to provide a further characterization of the dynamic response has been put in 135

the present analysis. With this aim, the wave amplitude of the 2LB is given the maximum

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of the values used in [42] and the unbalanced rotor operates under severe gravity residual conditions, thereby putting the stress on the concurring effects of the angular positions and bearing parameter f (the latter dimensionless parameter depends, among other quantities, on the lubricant viscosity). Besides to providing a characterization of the synchronous behaviour, 140

which confirms and extends previous results, a particular focus is put onto the 1/2-S, in the

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post-resonance region. In comparison to the response obtained with CB, the bifurcating behaviour of the subharmonic solution in the presence of 2LB shows a relatively rich spectrum of path structures, depending on the angular position and the bearing parameter. The observed

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path-pattern simplification, at higher values of the bearing parameter, indicates a more stable behaviour of the 1/2-S and represents an interesting feature presented by the study. The paper is organized as follows. Section 2 provides a brief description of the theoretical model, with hints to the continuation method and references to related literature. Section 3 presents the results of numerical investigation. In particular, subsection 3.1 reports data for the rotor on CB, operating with f = 0.02 (subsubsection 3.1.1) and f > 0.02 (subsubsection 3.1.2). Subsection 3.2 provides results for the rotor on 2LB, relative to the 1-S with f = 0.02

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(subsubsection 3.2.1), the 1/2-S with f = 0.02 (subsubsection 3.2.2), the 1-S with f > 0.02

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(subsubsection 3.2.3) and the 1/2-S with f > 0.02 (subsubsection 3.2.4). The analysis in the section is supported by eight tables of data. In the conclusive Section 4, the main aspects and

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features (effects) observed through the study are summarized and commented.

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2. Theoretical model

In order to evaluate the dynamic behaviour of the rotor on 2LB, it was decided to adopt

a relatively simple model, whose details were reported in [42]. The rotor is assumed rigid, 160

horizontally placed, on equal supports at its ends and statically unbalanced, thereby assuring the system symmetry with respect to the rotor middle plane. The analysis is so restricted to one half of the system, whose motion is given by the following equations, where u 1 and u 2 are

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the dimensionless coordinates of the journal center in a fixed frame of reference:

 0 u1 = u3        u02 = u4

u4 =

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−2q Ω u3 −2q Ω u4

1 u Ω2 1

− −

1 (u2 Ω2

+

f f 4λ2 Ω SFx

− u2,s ) +

(1) + U cos τ

f f 4λ2 Ω SFy

+ U sin τ

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  u03 =       0

The SFD is equipped with retaining springs and the quality of the journal centring in static conditions is expressed by the gravity residual u 2,s . The damper operates with oil supplied

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at ambient pressure, via a central, circumferential groove, free discharge at both ends and has a whole λ = L/D = 0.25 ratio. Three different angular frequencies, ω B , ω R and ω g (see Nomenclature) turn out to affect the system. The two latter frequencies are equal to 200 and 221.47 rad/s, respectively. The frequency ω B depends on the damper parameters and, for

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fixed geometry, on the oil viscosity. The ratio f = ω B /ω R is an important parameter to the analysis, together with the dimensionless speed Ω = ω/ω R . In particular, the f values 0.02, 0.03, 0.04 and 0.05 have been suitably selected, taking into account some literature results mentioned in [36]. The dimensionless unbalance U and gravity residual u 2,s have been fixed in 175

the present work to 0.3 and -0.7, respectively. In [36] it was shown that the above unbalance

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is in accordance with the ISO unbalance quality grade G=2.5, as far as the rotor speed is below 400 r.p.m., i.e. Ω = 0.209. In regard, it is worth observing that an Ω interval up to 7

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is assumed in the paper, far beyond the limit of 0.209. u 2,s = -0.7 indicates poor centering of the journal, i.e. a condition that favours a rich dynamics ([36]). The damping forces f SFx and 180

f SFy are obtained through numerical integration of the Reynolds equation specialized for a

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Finite Differences model, assuming laminar, isoviscous flow and Gumbel cavitation condition (see [36] for details relative to the mesh sizing and the solving procedure). The film thickness

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h was suitably determined from case to case, depending on the selected bore profile, according to the following dimensionless expression:

h = 1 − u1 cosδ − u2 sinδ + B cos(2δ + ϕ)

(2)

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The circular bearing was a particular case of the more general 2LB, obtained for wave amplitude B = 0. In the other cases B was assumed equal to 0.2 and the angular position ϕ was fixed in the set Φ ≡ {±3π/4, ±π/2, ±π/4, 0}. Fig. 1a shows how the bearing clearance is determined by the bore parameters, while Fig. 1b illustrates three examples obtained with B 190

= 0.2 and, respectively, ϕ = π/2, −π/4 and −3π/4.

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The system dynamics is examined with recourse to bifurcation theory and continuation method. A discrete model, equivalent to the continuous one of Eq.(1), is obtained in terms of Poincaré map, thanks to a robust and accurate ODE solver. By means of AUTO 97 [43] , the continuation of periodic solutions for the original system is thus reduced to the con195

tinuation of the corresponding fixed points. Since the eigenvalues of the Jacobian matrix of

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the Poincaré map are equal to the Floquet multipliers of the periodic orbits, the software can tackle the branching task needed for the continuation of periodic solutions, while assessing their stable/unstable character. The above procedure was also adopted in [44], where it was explained with more details, while a discussion with related example can be found in [45].

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200

(a)

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Fig. 1. (a) comparison of bearing clearances for the CB and the 2LB δL = - ϕ/2 . (b) examples of bearing clearances with ϕ = π/2, −π/4 and −3π/4.

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3. Results of numerical investigation 3.1. The circular bearing case 3.1.1. Results with f = 0.02

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Fig. 2(a) shows the behaviour of the synchronous solution (1-S), obtained through

numerical continuation, for the circular bearing case with f = 0.02. Stable and unstable solutions, in terms of the u 1 coordinate of the journal center vs. the rotor speed, are respectively represented by continuous and dotted lines. For the same conditions, Fig. 2(b) depicts a resonance curve that reports the maximum radius 210

of the journal center orbits vs. the rotor speed. This second diagram shows the

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only stable behaviour, because the orbits necessary to the amplitude evaluation are obtained through a brute-force integration of the system equations, carried out along

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the corresponding stable branches of Fig. 2(a).

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(a)

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Fig. 2. CB, 1-S, f = 0.02: (a) stable ( _ ) and unstable ( ... ) branches with bifurcation points. (b) maximum orbit radius evaluated along stable branches.

A continuous, upper branch a of the 1-S appears in both Figs. 2(a) and (b) 215

and represents stable orbits whose sizes increase steeply when Ω is raised from the

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lowest speed until resonance. The orbits attain a maximum radius at about Ω = 1.5 and maintain very large sizes in the subsequent interval of speed. This branch is

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indicated below by the bracketed sequence of three symbols: {<_>}, which indicate, respectively: lower bound of the speed-interval, stable behaviour, upper bound of the speed-interval.

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A further, lower branch of the 1-S, is represented in Fig. 2(a) and is marked by some bifurcation points (BPs), whose labels indicate the bifurcation character.

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This lower branch has a corresponding one in Fig. 2(b) that appears discontinuous, according to the observation reported above. The diagram clearly indicates that

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this branch represents post resonance solutions, whose orbits decrease in size with the rotor speed. Regarding the nature of the bifurcations, points S2, at Ω = 1.78, N1, at Ω = 2.30 and D2, at Ω = 4.20, denote respectively Fold, Neimark-Sacker

and Period Doubling bifurcations. The further BP, represented by N2/D4 in Fig. 2(a), indicates a transition from quasi-periodic to 1/2-subharmonic behaviour (or 230

vice-versa, when Ω is decreased), occurring at Ω = 2.82. This branch is indicated by means of the sequence {> ...S2_N1...N2/D4...D2_ >}, where the symbol ... denotes unstable solutions.

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The examination of the rotor behaviour under these conditions may be further carried out by means of Fig. 3(a), wherein a portion of the 1-S, excerpted from 235

Fig. 2(a), has been reported together with the 1/2-S. In order to accomplish a better readability, the 1/2-S and the 1-S have been plotted with thick and thin lines, respectively. The 1/2-S appears in figure with its two distinct branches, owing

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to the pair of u 1 values representing the solution at each given speed. For clarity purposes, the BPs relative to the 1/2-S have been marked with the respective labels 240

only along one of these two branches. Furthermore, in the same figure, some of the BPs already reported in Fig. 2(a), which are relative to the 1-S or in common to the 1-S and the 1/2-S, have been marked with bracketed labels. The double

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period solution emanates from the (N2/D4) point at Ω = 2.82, with an unstable branch (Fig. 3(a)). In this point, the character of the unstable 1-S modifies, owing to an exchange in the dominant Floquet multipliers: a pair of conjugate complex

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multipliers are replaced by a real negative multiplier in the outside position occupied with respect to the unit circle. The 1/2-S, after the turning point TP, undergoes three subsequent saddle-node bifurcations, alternating stability and instability, until it bifurcates into the synchronous solution at Ω = 4.25, point (D2). This path is 250

indicated by the sequence { (N2/D4)...TP...S1_S2...S3_(D2) }.

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A typical effect of the multiplicity, which is due to the presence of distinct saddlenode bifurcations, is illustrated in the example of Fig. 3(b), where two coexisting

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1/2-subharmonic orbits have been plotted for Ω = 4.0. The large orbit represents a solution taken on the S1_S2 branch, while the smaller one pertains to the S3_(D2) 255

segment. The hysteretic response that affects this speed region can be easily orig-

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inated, for instance, by decreasing the speed from a stable, synchronous operating condition on the 1-S branch, at a speed above Ω (D2) . In fact, lowering Ω beneath

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Ω (D2) , gives rise to small 1/2-subharmonic orbits, pertaining to the path S3_(D2), until a jump up to the branch S1_S2 occurs in S3, accompanied by the onset of a

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large 1/2-subharmonic motion of the journal center. On the other hand, increasing the speed in a run up from this condition on, would make the sub-synchronous motion suddenly change at S2 (Ω (S2) = 4.30 > Ω (D2) ), through a jump down, into the synchronous behaviour. It is also apparent, in the light of the above portrait of the different solution paths, that the sub-critical bifurcation at (N2/D4) corresponds to

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a hard generation of a 1/2-subharmonic, stable behaviour.

3.1.2. Results with f > 0.02

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(b)

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(a)

Fig. 3. CB, f = 0.02: (a) 1/2-S (thick lines) and 1-S (thin lines). (b) bearing clearance (...) with different orbits of the 1/2-S for Ω = 4.0, on the segment S3_(D2) ( ___ ) and on the segment S1_S2 ( _ _ _ ).

Giving the bearing parameter f a value of 0.05, in place of 0.02, determines substantial changes in the above results for the only synchronous behaviour. In 270

fact, the two branches in Fig. 2(a), i.e. the higher one {<_>} and the lower one

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{>...S2_N1...N2/D4...D2_>}, appear to be approximately split in two parts and recombined into two new different paths, namely {<_S1...S2_N1...N2/D4...D2_>}

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and {>_S3... >}, as shown in Fig.4(a). The resonant curve in Fig.4(b) clarifies, by comparison, the meaning of the different, stable branches, providing information about the orbit sizes. It is worth observing that:

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- The described reorganization of the branches (effect 1 ), as a result of increasing f, is confirmed by the data obtained at f = 0.03 and 0.04 and reported in Figs. 4(c)

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and (d), respectively. From comparison of Figs. 4(a),(c) and (d), the progressive shift of the branch {>_S3... >} towards higher speed can be inferred (effect 2 ).

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It is worth remarking that this branch represents very large motions. It must be observed too that the said effect 2 manifests on condition that effect 1 has already occurred. A progressive contraction of the unstable S1...S2 segment, as a further consequence of increasing f, may also be noticed (effect 3 ).

- This play of the bifurcating paths, which manifests as a branch-reorganization, is 285

very likely a result of the present bi-dimensional representation. In this regard, the more general portrait offered by the hyper-surface representing the solutions in a higher dimensional space, with f and further bifurcation parameters in

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addition to Ω, would deserve an insight. It is also worth remarking that the above said branch-reorganization, due to a change in the value of the bearing 290

parameter, is a well-known feature of the frequency responses for rigid rotors on SFD (with CB), under similar conditions. Examples are given in [46, 47], where the behaviour of resonance curves are analyzed adopting different parameters

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like unbalance, pressurization and bearing parameter.

(b)

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(a)

(d)

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(c)

Fig. 4. CB, 1-S: (a) solution curves, f = 0.05. (b) maximum orbit radius, f = 0.05. (c) solution curves, f = 0.03. (d) solution curves, f = 0.04.

Fig. 5 shows the behaviour of the 1/2-S, together with portion of the synchronous

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solution, for f = 0.05. In the diagram, the analogies to the results presented in Fig. 3(a) for f = 0.02 may easily be noticed. The 1/2-S branch {(N2/D4)...TP...S1_S2...S3_(D2)} originates in the bifurcation point (N2/D4), where the unstable synchronous solution bifurcates into an unstable 1/2-S. The solution path is similar to that of Fig.3(a), although with a shorter, about halved S1_S2 segment. Here Ω (S2) ' Ω (S3) , with Ω (S2) (= 3.49)

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just higher than Ω (S3) (= 3.45) and relatively distant from (D2). In consequence of these

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changes the hysteresis behaviour is less significant than that affecting the f = 0.02 case.

Fig. 5. CB, f = 0.05: 1/2-S (thick lines) and 1-S (thin lines).

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3.2. Effect of replacing the CB with the 2LB

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3.2.1. 2LB synchronous solution with f = 0.02

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An example of the changes that occur in the synchronous response for f = 0.02, when the CB is replaced by a 2LB, is presented in Figs. 6(a) and (b). The plotted bifurcation and resonance curves refer to a π-displaced bearing. The response differs from that of Figs. 2(a) and (b). The two branches {<_S1...S2_D1...D2_>} and

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{>...S3_>} resemble in fact the behaviour obtained with the CB operating under higher values of the bearing parameter and illustrated in Figs. 4(a),(c) and (d). In

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particular, it may be stressed that: - The adopted 2LB (ϕ = π) makes the Ω S3 threshold speed (where a large orbit

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solution can manifest) comparable to the one observed in Fig. 4(c), where the CB operates under f = 0.03.

- The Neimark-Sacker bifurcation, which occurs in the whole cases of Fig. 4, at the beginning of the unstable branch in the post-resonance region, is absent in the response of Fig. 6(a) (effect 4 ): the loss of stability of the 1-S is entirely due 320

to period doubling bifurcations (points D1 and D2).

13

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(b)

SC

(a)

Fig. 6. 2LB, 1-S, f = 0.02: (a) stable ( _ ) and unstable ( ... ) branches with bifurcation points. (b) maximum orbit

M AN U

radius evaluated along stable branches.

Further orientation angles of the 2LB, give rise to bifurcation diagrams that resemble the diagrams in Fig. 2(a) or Fig. 4(a), as in the previous case. Three examples are illustrated in Figs. 7(a)-(c), where ϕ is equal to −π/2, π/4 and π/2, 325

respectively. The behaviour in Fig. 7(b) appears to be midway between those of Figs. 7(a) and (c). Anyway, an insight into the same plots makes it possible to infer that the 1-S shows features and characteristics that are distinctive of each case. For

D

instance, the split of the upper branch (i.e. the said effect 1 ), observed and com-

330

TE

mented about Figs. 4(a)-(d), where f > 0.02, occurs here, for f = 0.02, on condition that the 2LB is given a suitable orientation. Nevertheless, the whole 2LB results obtained when f = 0.02 and ϕ ∈ Φ, here reported with restriction to the cases in

EP

Figs. 7(a)-(c) to save space, point out that the only orientation ϕ = π/2 yields a behaviour with continuous, not split upper branch (see Fig. 7(c)). The further effect 2, noticed for the CB at f increases, is already present here, when f = 0.02, even though at a smoothed degree, depending on ϕ.

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335

More generally, the different values ϕ ∈ Φ give respectively rise to branching

behaviours that differ in the lengths of the stable and unstable segments, as well as in the number and types of bifurcations. Tables 1 to 3 synthetically show the main characteristics of the said solutions. In particular, Table 1 reports the different

340

sequences that are distinctive of each branch. Table 2 shows the values assumed by the dimensionless rotor speed in correspondence of the different BPs. Table 3 gives the lengths of the speed intervals (LOSI) comprised between the said BPs, in order to allow further comparisons from the one behaviour to the other.

14

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(b)

D

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SC

(a)

(c)

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Fig. 7. 2LB, 1-S, f = 0.02, stable ( _ ) and unstable ( ... ) branches with bifurcation points: (a) ϕ = −π/2, (b) ϕ = π/4, (c) ϕ = π/2.

345

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Data in Table 1 make it possible to highlight some similarities of the CB response depicted in Fig. 2a to the 2LB behaviour for ϕ = π/2 and illustrated in Fig. 7c.

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Nevertheless, in the latter case, the stable solution that originates from the saddle S2, undergoes a period doubling, followed by a successive brief recover of stability {D3_N1}, before the torus bifurcation at N1. The subsequent branch is qualitatively the same in both cases, showing an unstable behaviour that ends in D2, where a

350

reverse period-doubling bifurcation occurs. Further data from Table 2 and Table 3 allow the following remarks to be made: - In the operation with CB, the whole speed interval interested by the postresonance instability starts at Ω = Ω N1 = 2.37 and stops at Ω = Ω D2 = 4.25, with a length given by (Table 3) w5b= Ω D2 −Ω N1 = 1.88. In particular, the length

355

of the sub-interval between Ω N1 and Ω N2/D4 , characterized by quasi-periodic

15

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Table 1 1-S, f = 0.02. Bifurcating paths and sequences of BPs.

>...S2_N1...N2/D4...D2_>

<_>

>...S2_D1...D3_N1...N2/D4...D2_>

<_>

−π/4

<_S1...S2_D1...D3_N1...N2/D4...D2_>

>...S3_>

π/4

<_S1...S2_D1...D3_N1...N2/D4...D2_>}

>...S3_>

−π/2

<_S1...S2_D1...D3/N1...N2/D4...D2_>}

>...S3_>

0

<_S1...S2_D1...D3/N1...N2/D4...D2_>}

>...S3_>

−3π/4

<_S1...S2_D1...D2_>

>...S3_>

3π/4

<_S1...S2_D1...D2_>

>...S3_>

π

<_S1...S2_D1...D2_>

>...S3_>

SC

π/2

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2LB

II path

RI PT

CB

I path

behaviour, is given by w4c = 0.45.

- When the rotor operates with π/2-2LB, the post-resonant, unstable speed-interval

D

turns out to be delimited by Ω D1 = 2.31 and Ω D2 = 4.91, with a length w2 = 2.6. The sub-interval wherein the rotor exhibit quasi-periodic behaviour, has a length w4c = 0.21 and is comprised between Ω N1 = 2.59 and Ω N2/D4 = 2.80.

TE

360

- Thus, from the above remarks, it may be observed that when the CB is replaced

EP

by the π/2-2LB, the post-resonance instability manifests in a larger speed interval (LOSI from 1.88 to 2.6). For convenience and further remarks, the whole segment representing this unstable behaviour is indicated as whole post-resonance unstable segment (WPRUS). The said lengthening of the WPRUS represents a

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365

feature hereafter denoted by effect 5. The comparison also shows that, along the WPRUS, the quasi-periodic behaviour originated from the torus bifurcation takes a shorter speed interval (LOSI from 0.45 to 0.21) when the CB is replaced by the π/2-2LB.

370

According to data in the same Table 1, a different, further subset of bifurcating curves, obtained for ϕ ∈ {−π/4, π/4, −π/2, 0} (subset A), can be put in evidence because of their path similarities. Two of these responses have been already depicted in Figs. 7(a) and (b). It is worth observing that:

16

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Table 2 1-S, f = 0.02. Dimensionless rotor speed at BPs.

Ω S2

Ω D1

Ω D3

Ω N1

Ω D3/N1

Ω N2/D4

Ω D2

Ω S3

−

1.76

−

−

2.37

−

2.82

4.25

−

π/2

−

1.81

2.31

2.44

2.59

−

−π/4

2.57

1.53

2.17

2.48

2.58

−

π/4

2.75

1.61

2.25

2.47

2.60

−

−π/2

2.56

1.63

2.19

−

−

2.46

0

2.57

1.51

2.17

−

−

2.62

−3π/4

2.38

1.73

2.31

−

−

3π/4

2.42

1.83

2.29

−

−

π

2.31

1.77

2.28

375

4.91

−

2.88

5.51

3.12

2.87

5.45

2.86

2.82

4.94

3.03

2.99

5.66

3.17

−

−

4.10

3.18

−

−

3.98

3.09

−

3.34

3.23

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2LB

2.80

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CB

RI PT

Ω S1

−

−

−

- Within the speed interval between S2 and D2, the -π/4 and π/4 solutions qualitatively behave in the same way as in the previously illustrated π/2 case (Table 1).

D

- The remaining two positioning angles of subset A, i.e. ϕ = −π/2 and 0, exhibit sequences that are slightly different from the previous two cases, owing to the absence of the short, stable segments D3_N1. Here, in fact, these two BPs

TE

380

collapse together to form the D3/N1 BP, where an exchange in the dominating

EP

Floquet multipliers occurs. - Table 2 also shows that the speeds Ω D1 and Ω D2 obtained from case to case, varying ϕ in this subset A, turn out to be respectively lower and higher than in the remaining other cases of the table. Consequently, the subset A-positions

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385

yield relatively long unstable branches between D1 and D2, as confirmed by the w2 index in Table 3, which attains the values of 3.49 and 3.34 for ϕ = 0 and −π/4, respectively. It is worth comparing these lengths too, as for the examined case with ϕ = π/2, to the value of 1.88 relative to the WPRUS in the operation

390

with CB. According to these remarks, the said effect 5 is also shared by the behaviours with ϕ ∈ subset A. - A further comparison regards the segments that represent unstable synchronous behaviour originated from N-S BPs. The index w4c, for ϕ = −π/4 and π/4, and w5c, for ϕ = −π/2 and 0, indicates that shorter (though comparable) LOSI are

17

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Table 3 1-S, f = 0.02. Characteristic LOSI (lengths of the speed intervals comprised between BPs).

w2

w3

w4a

w4b

w4c

w4d

w5a

w5b

w5c

w5d

−

−

−

−

−

0.45

1.43

−

1.88

−

−

π/2

−

2.6

−

0.13

0.15

0.21

2.11

−

2.32

−

−

−π/4

1.04

3.34

0.55

0.31

0.1

0.3

2.63

−

2.93

−

−

π/4

1.14

3.2

0.11

0.22

0.13

0.27

2.58

−

2.85

−

−

−π/2

0.93

2.75

0.47

−

−

−

2.12

0.27

−

0.36

2.48

0

1.06

3.49

0.60

−

−

−

2.67

0.45

−

0.37

3.04

−3π/4

0.65

1.79

0.8

−

−

−

−

−

−

−

−

3π/4

0.59

1.69

0.67

−

−

−

−

−

−

−

−

π

0.54

1.06

0.92

−

−

−

−

−

−

−

−

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2LB

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CB

RI PT

w1

Note: w1 = Ω S1 −Ω S2 , w2 = Ω D2 −Ω D1 , w3 = Ω S3 −Ω S1 , w4a= Ω D3 −Ω D1 , w4b= Ω N1 −Ω D3 , w4c= Ω N2/D4 −Ω N1 , w4d= Ω D2 −Ω N2/D4 , w5a= Ω D3/N1 −Ω D1 , w5b= Ω D2 −Ω N1 , w5c= Ω N2/D4 −Ω D3/N1 , w5d= Ω D2 −Ω D3/N1

obtained when the CB is replaced by the 2LB with the subset A angles.

D

395

Still, according to the data reported in Table 1, a remaining set of bifurcation

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curves, obtained for ϕ ∈ {−3π/4, 3π/4, π} (subset B), turn out to be characterized by comparable behaviours, about which the following aspects may be highlighted: - The instability loss of the 1-S in the post-resonance region, downward the saddle

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400

S2, is entirely due to period-doubling. This feature has been noticed just above,

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commenting the behaviour in Fig. 6(a), and has been indicated as effect 4. The further modification in the LOSI relative to the WPRUS, with respect to CB case, is exhibited when ϕ ∈ subset B. Nevertheless, differently from the previous

405

cases (effect 5 ) the LOSI takes quite low values. This feature, identified as effect 6, can be assessed by comparing the w2 index, from case to case, and is particularly evident for the π orientation, which yields the lowest value (1.06). When ϕ ∈ subset B, in fact, the instability loss and recovery of the 1-S are respectively delayed and anticipated (on run-up) with respect to the other cases,

410

as confirmed by the values of Ω D1 and Ω D2 reported in Table 2. - A further comparison between the subset A and the subset B is worth of remark

18

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and concerns the unstable behaviour between the two saddle nodes S1 and S2 (the CB and the π/2-2LB are excluded from this comparison, because of their un-split upper branches). The LOSI obtained for this unstable segment of the 415

synchronous solution is remarkably low when ϕ ∈ subset B. This effect (effect 3 ) is confirmed by the w1 values in Table 3 and is associated to a decrease

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(increase) in the values of Ω S1 (Ω S2 ). Consequently, the hysteretic behaviour and the jumps that are due to presence of the saddle-nodes S1 and S2, affect a speed range that is relatively short when ϕ ∈ subset B. 420

- The above remarks relative to the subset B show, in comparison to the other

SC

cases, a sort of softening of the unstable behaviour. This effect is represented by the reduction in the number of the bifurcation points, the disappearance of the quasi periodic response and the shortening of the LOSI corresponding to the

425

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segment S1...S2 and to the WPRUS. These results seem very likely due to the about vertical positions of the bearing that are accomplished by this set of ϕ values. This comment is enforced by observing that the said effects are more marked when ϕ = π.

3.2.2. 2LB 1/2-subharmonic solution with f = 0.02

D

430

TE

As shown in the previous sections, the instability loss of the synchronous solution, in the speed range downward resonance, is due to period-doubling, except when adopting a circular bearing. Hence, a further insight into the consequences of replac-

435

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ing the circular bush of the damper with the lobed one, can be achieved through an inspection of the 1/2-S behaviour in the said range of speed.

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Figs. 8(a)-(h) show the responses obtained when ϕ ∈ Φ by means of the continuation algorithm. The meaning of the line styles and the way BPs have been labeled in the plots are the same as in Figs. 3(a) and 5. Similarly to the previous Tables, Tables 4, 5 and 6 report data that are relevant to the different responses obtained

440

through continuation of the 1/2-S. About these data, the following comments are worthy of note: - The specific orientation given to the lobed bore significantly influences the complexity of the 1/2-S. The results obtained for ϕ ∈ {subset A, π/2, π} (Figs. 8(a)-(e) and (h)), characteristically exhibit two distinct paths. This 2-path,

445

branching behaviour (effect 7 ) is qualitatively different from the single-path re-

19

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Table 4 1/2-S, f = 0.02. Bifurcating paths and sequences of BPs.

II path

(N2/D4)...TP...S1_S2...S3_(D2)

-

π/2

(D1)_S1...(D3)

(N2/D4)...N1_N2...N3_S2...S3_(D2)

−π/4

(D1)_S1...(D3)

(N2/D4)...N1_S2...S3_D1...D2_(D2)

π/4

(D1)_S1...(D3)

(N2/D4)...N1_S2...S3_D1...D2_(D2)

−π/2

(D1)_S1...(D3/N1)

(N2/D4)...N1_S2...S3_(D2)

0

(D1)_S1_(D3/N1)

(N2/D4)...N1_S2...S3/D1...D2_(D2)

−3π/4

(D1)_S1...S2_S3...S4_(D2)

-

3π/4

(D1)_D1...D2_S1...S2_S3...S4_(D2)

π

(D1)_D1...D2_>

SC

2LB

-

S1_S2...S1

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CB

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I path

sponses obtained when ϕ = ±3π/4 (Figs. 8(f) and (g)). It is worth observing that also the 1/2-S of the CB has a single-path pattern (Fig. 3(a)). - In the presence of the 2-path behaviour and ϕ ∈ subset A or ϕ = π/2, the first path is attached to the 1-S through a (D1) BP, at lower speed, and a (D3) (Figs. 450

8(a)-(c)) or a (D3/N1) BP (Figs. 8(d) and (e)), not far distant, at a higher

D

speed. This path consists of a simple sequence of two branches, one stable and the other unstable, separated by a saddle node S1. The LOSI relative to the

TE

stable behaviour between Ω (D1) and Ω S1 is expressed by the k1 index in Table 6, which attains the maximum (0.89) and the minimum (0.28) values for ϕ = π/2 455

and π/4, respectively. The second of the two paths is delimited by (N2/D4) and

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(D2) BPs, both placed along the unstable synchronous branch, and exhibits, from case to case, comparable LOSI, as confirmed by the index k2 = Ω (D2) -

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Ω (N2/D4) in Table 6. Along this second path, a stable 1/2-S sets in after an initial unstable branch between (N2/D4) and N1. The stable behaviour changes

460

further on, owing to the occurrence of quasi-periodic behaviour (N2...N3 segment for ϕ = π/2) or period-doubling (D1...D2 segments for ϕ = 0, ±π/4) and fold bifurcations (S2...S3 segments). In particular, when ϕ = −π/2, the occurrence of saddle-nodes represents the only bifurcation type which affects the stable 1/2-S over the path from N1 to (D2). When ϕ = 0, the unstable branch between the

465

two folds directly undergoes a period doubling in S3/D1. The different folds entail the presence of coexisting, different solutions at a given speed, respectively pertaining to distinct, stable segments. In that same

20

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(b)

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SC

(a)

(d)

EP

TE

D

(c)

(f)

(g)

(h)

AC C

(e)

Fig. 8. 2LB, f = 0.02: 1-S (thin lines) and 1/2-S (thick lines). Bifurcation points labeled within brackets refer to 1-S or are in common to the two solutions.

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Table 5 1/2-S, f = 0.02. Dimensionless rotor speed at BPs.

CB

2LB π/2

−π/4

π/4

−π/2

0

−3π/4

3π/4

π

2.31

2.29

2.28

-

2.31

2.17

2.25

2.19

2.17

Ω (D2)

4.25

4.91

5.51

5.45

4.94

5.66

4.10

3.98

3.34

Ω (D3)

-

2.44

2.48

2.47

-

-

-

-

-

Ω S1

2.60

3.20

2.82

2.53

3.02

2.64

3.41

4.58

-

Ω S2

4.31

4.47

4.34

4.50

4.28

4.41

3.28

3.51

-

Ω S3

3.38

4.14

4.27

4.32

4.09

-

4.72

3.84

-

Ω S4

-

-

-

-

-

-

3.70

3.76

-

Ω N1

-

2.97

3.06

3.67

3.06

3.07

-

-

-

Ω N2

-

3.32

-

-

-

-

3.70

3.76

-

Ω N3

-

4.33

-

-

-

-

3.70

3.76

-

Ω D1

-

-

4.39

4.40

-

-

-

2.81

3.05

Ω D2

-

-

5.21

5.15

-

5.26

-

3.56

3.74

Ω (D3/N1)

-

-

-

-

2.45

2.49

-

-

-

Ω (N2/D4)

2.82

2.81

2.89

2.89

2.82

2.92

-

-

-

Ω (S3/D1)

-

-

-

-

-

4.36

-

-

-

TE

D

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SC

RI PT

Ω (D1)

speed, further possible, coexisting motions, with periodic (for instance 1/3-

470

EP

subharmonic, as in Fig. 14(a) of [36]) or non-periodic nature (for instance chaotic, as in the next Fig. 9(b)) would deserve a specific search. An example of coexisting motions is provided with reference to the π/2 orientation. In this case,

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a 1-S, a large 1/2-S or a small 1/2-S may occur within the interval [Ω (D1) , Ω S1 ] (Fig. 8(a)), thereby justifying the possible onset of jumps. Fig. 9(a) depicts the orbits obtained for the 1/2-subharmonic behaviours and the synchronous

475

response when Ω = 3.1, adopting different initial conditions. The same figure also shows, at smaller scale, the Poincaré sections relative to the quasi-periodic motion into which the small 1/2-S bifurcates at N2, here evaluated for Ω = 3.7.

- Choosing ϕ = π (Fig. 8(h)) gives still rise to a double path 1/2-S. The first branch {(D1)_D1...D2_>} emanates from (D1), as in the previous cases, with 480

orbits that are rapidly growing in size with the rotor speed, until it passes through a period doubling at D1. A subsequent reverse flip bifurcation, in the point D2,

22

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Table 6 1/2-S, f = 0.02. Characteristic LOSI (lengths of the speed intervals comprised between BPs).

CB

2LB −π/4

π/4

−π/2

0

−3π/4

3π/4

π

1.29

-

k1

-

0.89

0.65

0.28

0.83

0.47

1.1

k2

-

2.10

2.62

2.56

2.12

2.74

k3

-

2.60

3.34

3.20

2.75

k4

-

-

-

-

k5

1.65

-

-

k6

1.88

-

-

RI PT

π/2

-

-

3.49

1.79

1.69

1.06

-

-

2.41

-

-

-

-

-

-

-

-

-

-

-

-

-

-

SC

-

Note 1: k1 = Ω S1 −Ω (D1) , k2 = Ω (D2) −Ω N2/D4 , k3 = Ω (D2) −Ω (D1) , k4 = Ω S3 −Ω (D1) ,

M AN U

k5 = Ω (D2) −Ω S1 ,k6 = Ω (D2) −Ω (N1) .

Note 2: The only relevant k indexes have been evaluated with the CB set-up. The indexes k3 and k6 are the same as w2 and w5b in Tab. 3, respectively.

re-establishes the 1/2-S, which survives beyond the upper limit of the considered speed range (until about Ω = 10). The second 1/2-S path consists of an isolated branch {S1_S2...S1}, delimited by the two saddle-nodes S1 and S2 and characterized by orbits that are smaller than those on the previous path. Fig.

D

485

TE

9(b) shows two orbits pertaining to the two paths and obtained at Ω = 4.5. In this same speed, the above two 1/2-S coexist with two, different, synchronous motions (according to the respective branches in Fig. 8(h)) and a chaotic be-

490

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haviour, whose attractor is depicted together with the bearing clearance at a smaller scale, in the bottom left corner of Fig. 9b.

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- When the bearing is positioned with ϕ = ±3π/4 (Figs. 8(f) and 8(g)), a singlepath, bifurcating behaviour characterizes the 1/2-S, showing the presence of segments that have been already observed in the above two-path responses. Here the 1/2-S emanates from (D1) and undergoes, while twisting, the different saddle-

495

node bifurcations S1, S2, S3 and S4, until (D2). The LOSI relative to the whole path for ϕ = −3π/4 and ϕ = 3π/4, are respectively given by k4 = Ω S3 - Ω (D1) =

2.41 and k1 = Ω S1 - Ω (D1) = 2.29 (see Table 6, Fig. 8(f) and (g)). These LOSI are shorter than the analogous ones obtained in the diagrams of (Figs. 8(a)-(e)), where k3 is calculated as Ω (D2) ) - Ω (D1) and attains its maximum 3.49 for ϕ = 0. 500

It must be pointed out that the index k3 coincides with w2 in Table 3 and is repeated here for the sake of comparison.

23

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- The reduction from two to one in the number of paths and the shortening of the LOSI achieved through ϕ = ±3π/4, make the resulting behaviour of the 1/2-S comparable to that obtained with CB and depicted in Fig. 3(a). As a matter of fact, with respect to the latter case, neither a decrease in the number of the BPs

505

that occur in turn along the solution branches, nor a significant shortening of the

RI PT

LOSI are obtained with the 2LB mount. Compare, on the one hand, the organization of the respective paths shown in Table 4. On the other hand, in Table 6, compare k4 and k1, respectively evaluated for ϕ = −3π/4 and 3π/4, to the indexes k5 and k6 relative to the CB. Nevertheless, the complete disappearance

510

of Neimark-Sacker bifurcations from the same 2LB responses (effect 8 ) is worthy

SC

of remark. This feature is also present in the response obtained for ϕ = π. A trace of the Neimark-Sacker bifurcation that characterizes the instability loss of

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the CB synchronous solution, is maintained within the two-path patterns of the 1/2-S for ϕ ∈ (π/2, subset A).

AC C

EP

TE

D

515

(a)

(b)

Fig. 9. 2LB, f = 0.02, multiple solutions, (... : clearance border): (a) ϕ = π/2, Ω = 3.1 (smaller scale: Poincaré sections of quasi-periodic motion for Ω = 3.7). (b) ϕ = π, Ω = 4.5 (smaller scale: Poincaré sections of a chaotic solution).

3.2.3. 2LB synchronous solution with f > 0.02 Some examples of the synchronous behavior obtained for operation on 2LB, under f parameter equal to 0.03 and 0.04, are depicted in Figs. 10(a)-(f). Table 7 520

summarizes relevant data for f = 0.05.

24

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Figs. 10(a)-(c) depict the solutions determined with f = 0.03, adopting ϕ = −3π/4, π/2 and π, respectively. Analogously, Figs. 10(d)-(f) show the responses for f = 0.04, assigning ϕ = −3π/4, −π/2 and π, respectively. With regard to the

SC

RI PT

depicted behaviours and the reported data, the following comments may be made:

(b)

TE

D

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(a)

(d)

AC C

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(c)

(e)

(f)

Fig. 10. 2LB, 1-S, stable ( _ ) and unstable ( ... ) branches with bifurcation points: (a)-(c) f = 0.03, (d)-(f) f = 0.04.

525

- Comparing the plots in Fig. 4(c) and Figs. 10(a)-(c), on the one hand, and those in Fig. 4(d) and Figs. 10(d)-(f), on the other hand, it can be inferred that the

25

ACCEPTED MANUSCRIPT

adoption of lobe profile enforces the effect 2, making the branch {>_S3... >} shift upward, till to a complete disappearance from the speed interval. Yet, under higher f values, this effect is less dependent on the given angular orientation 530

(compare the plots in Figs. 7a-c obtained with 2LB and f = 0.02). It may be further observed that the adoption of the lobe profile and the increases in f are

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concurring in order to emphasize the effect (compare Figs. 10a-c to Figs. 10d-f). - Increasing f and giving suitable ϕ angles, the unstable branch S1...S2 contracts (effect 3 ), till to a complete disappearance (compare Figs. 10(a)-(f)). 535

- Regarding the WPRUS, data from Table 7 make it possible to ascertain the dif-

SC

ferent responses yielded by angular positions with ϕ ∈ (π/2, subset A), on the one hand, and ϕ ∈subset B on the other hand. In particular, the w2 values show that the former angular positions, despite the path simplification with respect to

540

M AN U

the operation under f = 0.02 (compare the number and types of BPs), maintain relatively large LOSI correspondent to the WPRUS. Nevertheless, a comparison to the corresponding values of w2 in Table 3 shows that increasing f from 0.02 to 0.05, a shortening of the correspondent LOSI (which turns out to be significant when ϕ = π/2, π/4 and 0) is obtained. Differently, in the cases represented by the subset B, the LOSI relative to the WPRUS are still rather low, though higher than the corresponding ones obtained for f = 0.02. Even under f = 0.05, the

D

545

lowest LOSI is achieved with a π orientation. The corresponding value of 1.16,

TE

expressed by w2, is slightly higher than 1.06 obtained for f = 0.02.

550

EP

3.2.4. 2LB 1/2-subharmonic solution with f > 0.02 Main findings relative to the 1/2-S behavior with 2LB, under higher values of the

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f parameter, are summarized here below, with reference to Figs. 11(a)-(h), where f = 0.03 and 0.05. Bifurcating paths and rotor speeds relative to the BPs in the plots and to the corresponding CB cases are respectively given in Table 8 and 9. The reported diagrams and the comparison with the corresponding results at f = 0.02,

555

show quite clearly that the higher f, the more simplified are the resulting bifurcating paths. As expected, this simplification, which is taken in the analysis as a further, distinctive effect (effect 9 ), is depending on the given bearing orientation. The results for ϕ = π/2, for instance (Figs. 11(a) and (b)), show that the relatively complex behavior observed at f = 0.02 (Fig. 8(a)) turns out to be decidedly softened when f

560

= 0.05 (Fig. 11(b)). In this latter case, in fact, not only a change from a two-path to a single-path branching occurs, but the complete stability of the 1/2-S is also gained.

26

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The complete stability is also obtained for f = 0.05 and ϕ = −3π/4 (Fig. 11(d)) and π/4 (not reported). Differently, only an incomplete stability is recovered by the 1/2-S when ϕ = 3π/4 (Figs. 11(e) and (f)), ϕ = π (Figs. 11(g) and (h)), ϕ = 0 (not reported, single-path similar to that in 11(b), with flip BPs) , −π/2 and −π/4

565

(not reported, double-path patterns, similar to that in 11(a), though with completely

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stable upper path). Contrast with these simplified patterns, the relatively involved structures (especially in some cases) depicted in Figs. 8(a)-(h). In this regard, it is worth remarking the relative stability of the bifurcating structure obtained for the CB mount, (Figs. 3(a) and 5), which exhibits similar patterns under the different

570

SC

values assigned to f.

Table 7

Ω N1

Ω D1

Ω D3

Ω D4

Ω D2

w5b

w2

<_S1...S2_N1...N2/D4...D2_>

2.55

-

-

-

4.20

1.65

-

π/2

<_D1...D2_>

-

3.12

-

-

4.86

-

1.74

−π/4

<_D1...D3_D4...D2_>

-

2.28

2.65

3.19

5.26

-

2.98

π/4

<_D1...D2_>

-

3.15

-

-

5.25

-

2.10

−π/2

<_D1...D3_D4...D2_>

-

2.38

2.72

3.0

4.87

-

2.49

0

<_D1...D2_>

-

3.17

-

-

5.36

-

2.19

CB

D

path

−3π/4

<_D1...D2_>

-

2.33

-

-

4.32

-

1.99

3π/4

<_D1...D2_>

-

2.50

-

-

4.07

-

1.57

π

<_D1...D2_>

-

2.44

-

-

3.60

-

1.16

TE

2LB

M AN U

1-S, f = 0.05. Bifurcating paths, rotor speed at BPs and characteristic LOSI.

EP

Note 1: The path relative to the CB is one of the two paths obtained in this case and illustrated in Fig. 4(a).

Table 8

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Note 2: w2 and w5b are defined in Table 3.

1/2-S, f > 0.02. Bifurcating paths for f = 0.03 and 0.05.

CB

π/2 2LB

f = 0.03

f = 0.05

(N2/D4)...TP...S1_S2...S3_(D2)

(N2/D4)...TP...S1_S2...S3_(D2)

(D1)_S1...(D3) (N2/D4)_N1...N2_S1...S2_(D2)

(D1)_(D2)

−3π/4

(D1)_S1...S2_(D2)

(D1)_(D2)}

3π/4

(D1)_S1...S2_S3...S4_(D2)

(D1)_S1...S2_(D2)

π

(D1)_S1...S2_(D2)

(D1)_S1...S2_(D2)

27

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ACCEPTED MANUSCRIPT

(b)

M AN U

SC

(a)

(d)

EP

TE

D

(c)

(f)

AC C

(e)

(g)

(h)

Fig. 11. 2LB, 1/2-S. Comparison of solutions with f = 0.03 and 0.05: (a),(b) ϕ = π/2. (c),(d) ϕ = −3π/4. (e),(f) ϕ = 3π/4. (g),(h) ϕ = π.

28

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Table 9 1/2-S, f > 0.02. Dimensionless rotor speed at BPs for f = 0.03 and 0.05.

f = 0.03

π/2

−3π/4

3π/4

π

CB

2LB π/2

−3π/4

3π/4

RI PT

2LB

π

-

2.41

2.32

2.37

2.33

-

3.12

2.44

2.52

2.46

Ω (D2)

4.22

4.80

4.16

4.02

2.44

4.21

4.75

4.28

4.05

3.62

Ω (D3)

-

2.53

-

-

-

-

-

-

-

-

Ω N1

2.55

2.69

-

-

-

-

-

-

-

-

Ω S1

2.71

2.64

4.29

3.68

3.75

2.94

-

-

3.44

3.53

Ω S2

3.87

4.21

3.82

3.47

3.39

.49

-

-

3.44

3.50

Ω S3

3.43

4.34

-

3.82

-

3.45

-

-

-

-

Ω S4

-

-

-

Ω N1

-

3.40

-

Ω N2

-

4.10

-

Ω (N2/D1)

-

-

-

Ω (N2/D4)

2.90

2.90

-

3.79

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

2.90

-

-

-

-

-

-

-

-

-

-

-

D

4. Conclusions

SC

Ω (D1)

M AN U

CB

f = 0.05

575

TE

A previous, theoretical study of the author was carried out about a rigid rotor on SFD equipped with two-lobe, wave bush. The use of numerical continuation made it possible to characterize efficiently the dynamic behaviour, with suitable account of the shape parameters

EP

of the lobed damper bush (i.e. wave amplitude and angular position). The obtained portrait of the system dynamics was mainly focused on the synchronous behaviour and necessarily restricted, given the number of parameters influencing the system. The present investigation was aimed at confirming and extending the previous results, through a further insight into the

AC C

580

effects of the lobe geometry, taking particularly into account the role of the bearing parameter f. The wave amplitude of the 2LB bore was kept fixed to the maximum value adopted in the previous investigations, while the angular position ϕ was varied in a suitable set of values. The comparison of the rotor behaviour on CB to the outcomes with the 2LB mount, made it 585

possible to highlight the following features: - effect 1 : reorganization in the bifurcating paths of the 1-S. This effect, consisting in a characteristic change in the path pattern, is achieved with CB for f > 0.02. In the presence of 2LB, suitable angular positions give rise to the said reorganization even when f = 0.02. Under higher f values, the reorganization appears to be achieved whatever is

29

ACCEPTED MANUSCRIPT

590

ϕ. - effect 2 : shift of the {> ...S3_>} branch towards higher speed. This behaviour, which manifests on condition that effect 1 has already occurred, emphasizes increasing f, when the rotor is mounted on CB. In the presence of 2LB and f = 0.02, it is favoured by

595

RI PT

suitable angular positions and is accentuated at higher f values. - effect 3 : contraction of the unstable S1...S2 segment of the 1-S. This effect may be observed in the CB responses increasing f. The replacement with 2LB accentuates the contraction, particularly when ϕ ∈ (subset B). Higher f values determine a complete

SC

disappearance of the segment.

- effect 4 : disappearance of the torus bifurcations in the post-resonance branches of the 600

1-S. Differently from the CB responses, which maintain the presence of these bifurcations

M AN U

whatever is f, the lobe profile makes the N-S BPs completely vanish when f >= 0.04. - effect 5 : WPRUS (1-S) lengthening. This effect is observed comparing the synchronous behaviours of the 2LB with ϕ ∈ (π/2, subset A) to the CB responses. This effect is more moderate increasing f. 605

- effect 6 : WPRUS (1-S) contraction. This behaviour may be noticed when ϕ ∈ ( subset with ϕ = π and f = 0.02.

D

B). Raising f makes the effect more moderate. The lowest LOSI, equal to 1.06, is obtained

TE

- effect 7 : reorganization in the bifurcating paths of the 1/2-S, in the presence of 2LB. With suitable angles, a two-path branching characterizes the 1/2-S. This behaviour contrasts 610

with the single-path one observed, whatever is f, in the CB responses and is maintained

EP

by the 2LB mount even at higher f values, for some positioning angles. - effect 8 : disappearance of the torus bifurcations in the 1/2-S behaviour, in the presence

AC C

of 2LB. This feature is observed for f =0.02 when ϕ ∈ (subset B). The effect extends to further ϕ values when f is increased.

615

- effect 9 : simplification in the bifurcating paths of the 1/2-S, in the presence of the 2LB and higher f values. The path-patterns obtained for the 1/2-S, adopting the 1/2-S when f = 0.02, appear relatively involved in comparison to the 1/2-subharmonic behaviour observed with the CB and f = 0.03 or 0.05. Nevertheless, raising f to 0.05, the doublepath pattern vanishes in most cases, in favour of a single-path, with almost completely

620

(ϕ = 3π/4, 3π and 0) or completely stable (ϕ = π/2, −3π/4 and π/4) structure. Even when double-path patterns survive (ϕ = −π/2 and −π/4), the instability appears to be marginal. Hence, the claimed simplification substantially indicates a more stable

30

ACCEPTED MANUSCRIPT

behaviour gained by the 1/2-S in the operation with 2LB at higher f values. In summary, it may be concluded that the lobed geometry, as compared to the common 625

circular one, positively influences the stability of the synchronous solution, especially in case of about vertical angular positions of the bearing. Regarding the 1/2-subharmonic

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solution, the stability improvement due to the 2LB turns out to be noticeable increasing the bearing parameter and shows to be less dependent on the bearing orientation. These results may encourage further work based on enhancements of the theoretical model and experimental investigation.

SC

630

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635

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AC C

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760

35

ACCEPTED MANUSCRIPT Highlights

RI PT SC M AN U TE D



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 

The role of unconventional bore shapes on the operation of SFD is worthy of investigation. A two-lobe wave profile emphasizes the effects of the angular positioning of the damper bearing. The dynamics of an unbalanced rigid rotor on SFD with such bearings is theoretically studied. The analysis through numerical continuation focuses on synchronous and 1/2 subharmonic behaviour. Stability is enhanced, especially giving suitable orientations and raising the bearing parameter.

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 