Fractional modeling applied to tilting-pad journal bearings

International Journal of Dynamics and Control https://doi.org/10.1007/s40435-020-00656-5 Fractional modeling applied to tilting-pad journal bearings Carlos A. Valentim Jr.1 · José A. Rabi1 · Sergio A. David1 Received: 2 April 2020 / Revised: 13 May 2020 / Accepted: 6 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract Tilting-pad journal bearings can be found in a variety of industrial applications due to their stabilizing effects and damping behavior. Reliable mathematical modeling is demanded to analyze and control those dynamical systems. This paper adopts a fractional approach to extend an extant mathematical model of test rig for active tilting-pad bearings. Along with related results, the theoretical framework here presented may contribute to better understand such dynamic behavior, possibly supporting the development of more efficient control strategies for this important mechanical system. Keywords Fractional calculus · Bearings · Dynamic model · Vibrations 1 Introduction machines due to their damping characteristics and facilities in assembling rotor-bearing systems [1,2]. Designing high-efficiency rotating machinery as turbines, TPJB can be combined with a hydraulic chamber system pumps, and compressors is a fundamental step in produc- in order to apply active control techniques [3]. For control tion processes, since these parts must have high performance purposes a reliable mathematical model is necessary in order and continuous availability to minimize the risk of jeopar- to represent the dynamic behavior as close as possible to the dizing production flow. Moreover, rotor-dynamic analysis, real mechanical system. While the dynamic behavior of that stability-evaluation, and mainly the control of vibration lev- system has been continuously studied and modeled, these els in rotating machines remain a major concern for both approaches have their validity often argued when compared designers and users of these equipment. to experimental data [4], thus claiming for alternative and The maintenance of low vibration levels requires mech- more accurate mathematical modeling on nonlinear oscilla- anisms to dissipate vibration energy and hydrodynamic tions [5,6]. bearings can be efficiently used [1]. Tilting pad journal bear- Methodologies based on fractional modeling have obtained ings (TPJB) are a type of hydrodynamic bearing normally encouraging results supporting the design of a plethora of used in high-speed machinery, particularly in applications mechanical systems. Although TPJB have been studied for where plain cylindrical bearings might present problems many years [2,7–9], one can only find some similar fractional with self-induced vibrations. Regarding dynamic stability, systems in the literature [10–12] while fractional calculus TPJB’s properties make them outstanding in relation to their (FC) specifically applied to TPJB modeling remains poorly pairs/counterparts. They are also suitably used in turbo- explored. FC-based model could allow one to better adjust dynamic coefficients, significantly reduce vibration ampli- tudes of a rotor-bearing system, describe the complicated frequency-dependence of damping materials, and improve This study is financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance code 001. the understanding of various possible types of complex dynamic behaviors (e.g. period-doubling bifurcation, sud- B Sergio A. David den transition, and quasi-periodic from periodic motion to [email protected] chaos). This study adopts fractional calculus (FC) [13,14] Carlos A. Valentim Jr. for generalizing the TPJB mathematical model reported by [email protected] Santos [3]. FC is a branch of mathematics that that general- 1 Systems Dynamics Group, University of São Paulo, izes integer order calculus, in which derivatives and integrals Pirassununga Campus, Av. Duque de Caxias Norte 225, can assume non-integer and even complex orders [15]. Pirassununga, SP 13635-900, Brazil 123  C. A. Valentim Jr. et al. These perspectives motivated us to use FC to possibly piezoactuator and an output signal Y R from a displacement improve aforementioned TPJB mathematical model, spe- sensor for an active TPJB. The structure of this mechanical cially because FC has effectively described time-delayed system is shown in Fig. 1. chaotic systems [16], diffusive [17–19] and oscillatory phe- The mathematical model representing the dynamical nomena [20–26]. behavior of such system is originally a second-order differ- A model with non-integer orders can capture non-linear ential equation, whose generalization behavior that traditional models fail to represent. Besides, the arbitrary order arises as an extra best-fitting parameter, L2 AR allowing a more faithful representation of experimental data D 2α Y R (t) + d̄ yy D α Y R (t) IA [27,28].  2  L AR L2 AP Bearing in mind successful equipment design, scale-up + k̄ yy + k p Y R (t) = F(t), (1) and optimization, comprehensive RD&I (research, develop- IA IA ment and innovation) activities properly combine distinct solution methods (i.e. analytical, experimental and numer- is proposed so as to be analytically solved. In Eq. 1, Y R ical). Virtualization is a versatile engineering tool not only to is the rotor vertical displacement, D α and D 2α are opera- save material, energy and/or human resources in industrial tors of derivatives of arbitrary orders, I A is the lever-rotor RD&I [29] but also when obtaining experimental or real- moment of inertia, L A R and L A P are the lengths related operation data becomes awkward or even unfeasible [30]. to rotor and piezoactuator positions, k p and C p are stiff- In particular, analytical solutions are helpful to numerically ness and piezoelectric constants of the actuator, d̄ yy and validate simulators in idealized scenarios. k̄ yy are global damping and stiffness constants regarding Therefore, also considering the importance of an accurate the bearings, U p (t) is the input signal of the actuator, and mathematical model capable of validating experimental data F(t) = L A RI AL A P C p k p U p (t) is the result of the input signal on TPJB, in this paper we derive and propose an analytical U p (t). In [3], a 4-step experimental identification procedure solution for the generalized differential equation governing was carried out to determine the dynamic coefficients of Eq. 1 aforesaid system. by applying signal feedback control based on the relation of piezoactuator input signal U P and output signal Y R given by 2 Mathematical modeling a displacement sensor. Thus, in this brief communication, those parameters are allegedly known. 2.1 Approach using FC The system is then explored for non-integer orders within the range 0.5 < α ≤ 1. The arbitrary order α is considered Santos [3] proposed an identification procedure scheme an additional free parameter that can be best-fitted with help based on the relation between an input signal U p from a of optimization routines against available experimental data. Fig. 1 Santos’ [3] test rig and mechanical model 123 Fractional modeling applied to tilting-pad journal bearings This feature may provide the model given by Eq. 1 with Considering the definition in (2) and admitting that Y R (t) additional resources to describe the investigated scenario or can be written as a power series, one can use the property desired physical characteristics. The integer order differential [32] given by (7) to extend it from t k to a series of Cn t nα , so equation is recovered by setting α = 1 in Eq. 1. that 2.2 Fractional series solution of the generalized dα k Γ (1 + k) k−α t = t . (7) model dt α Γ (1 + k − α) There are different approaches to define the fractional deriva- Thus, one obtains the fractional power series for D α Y R (t) tive represented by the operator D α , each one having its own and D 2α Y R (t) given by (8) and (9), respectively, so that advantages and applications [13]. In this work, we use Caputo ∞  left-handed definition represented by Γ (nα + 1) D α Y R (t) = t (n−1)α , (8) Γ [(n − 1)α + 1] d n=1 D α Y R (t) = I 1−α Y R (t) ∞  Γ (nα + 1) dt D 2α Y R (t) = t (n−2)α .  t (9) 1 dY R (ξ ) Γ [(n − 2)α + 1] = (t − ξ )−α dξ. (2) n=2 Γ (1 − α) 0 dξ Next, one can insert (8) and (9) into (4) to obtain Per definition, Caputo’s derivative operator composes a traditional derivative followed by a fractional Riemann– ∞  Γ (nα + 1) Liouville integral of order α − 1. Therefore, if the operator Cn t (n−2)α is applied to any constant, the result will be zero since the Γ [(n − 2)α + 1] n=2 traditional derivative will be applied first. As a result, such ∞  Γ (nα + 1) definition can treat boundary conditions in a similar way as + A(ω) Cn t (n−1)α (10) Γ [(n − 1)α + 1] integer order problems, which is convenient when modeling n=1 ∞ physical phenomena. In Eq. (2), Γ is gamma function, I is an integral operator and ξ is a dummy variable. + B(ω) Cn t nα = F(t). n=0 A power series method based on fractional Taylor series [31] is adopted to find a solution for (1). Using this approach, After writing every term of the equation under the same the rotor vertical displacement Y R (t) is represented as a frac- summation symbol and adjusting the indexes one obtains, tional power series centered at x0 = 0, that is, ∞   ∞  Γ [(n + 2)α + 1] nα Y R (t) = Cn t nα . Cn+2 t (3) Γ (nα + 1) n=0 n=0 Γ [(n + 1)α + 1] nα +A(ω) Cn+1 t Equation (1) is represented in a shorter form, Γ (nα + 1) ∞     (i ω)nα D 2α Y R (t) + A(ω)D α Y R (t) + B(ω)Y R (t) = F(t), (4) +B(ω)Cn t nα = F0 F̄ p t nα . (11) Γ (nα + 1) n=0 where Hence, a recurrence equation can be found for a non-trivial L2 solution by rearranging (11), so that the coefficients of the AR A(ω) = d̄ yy , (5) fractional power series are given by IA and F0 F̄ p (i ω)nα − A(ω)Cn+1 Γ [(n + 1)α + 1)] − B(ω)Cn Γ (nα + 1) Cn+2 = . Γ [(n + 2)α + 1]   L2 AR L2 AP (12) B(ω) = k̄ yy + kp , (6) IA IA Considering the power series that represents Y R (t), given A(ω) and B(ω) depend on frequency ω and also on param- by (3), the coefficients C0 and C1 remain unknown. They can eters shown in (5) and (6), while F(t) is related to the be set by the boundary conditions of the model. Firstly, we excitation force over the rotor-lever system depending on evaluate the initial displacement Y R0 at x = 0 in (13) and parameters mentioned in (1) besides time t. define the coefficient C0 in (14), namely, 123  C. A. Valentim Jr. et al. ∞  6. 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