The unbalanced rotating cylinder partially filled with fluid; multiple scales analysis of a forced Korteweg–de Vries–Burgers equation

The paper is concerned with an unbalanced cylindrical rotor containing a small amount of fluid, spun out to form a thin layer on the inner surface of the cylinder. The main interest is in the possibility of this fluid layer to counterbalance an unbalanced point mass. By such an application, the system is often called a ‘fluid balancer.’ The paper considers the case where the fluid is not locked-in to the forcing frequency dictated by the unbalanced mass, but is, with the rotor, in a state of asynchronous whirl. This will imply an inherent slow drift away from a balanced condition. Another main interest of the paper is the derivation of approximate analytical solutions to the nonlinear, forced equation that governs the fluid layer thickness perturbation. This equation is of the forced Korteweg–de Vries–Burgers type, and the analysis is based on the method of multiple scales. The leading-order solution is capable of giving a qualitative explanation of the balancing effect of the fluid, in other words, to explain the mechanics of the fluid balancer. Good agreement between theoretical and experimental results is found.


Introduction
The dynamics and stability of rotating machinery has been of concern for more than a century [1][2][3][4]. Setting up the equations of motion for an unbalanced rotating disk on an elastic shaft, one will find that this is a forced vibration problem [5], with the forcing being due to the centrifugal force of the unbalanced mass. (The shaft can possibly be represented just by a pair of springs mounted to the rotor in perpendicular directions. This is the approach taken in the present paper.) When the angular velocity is large enough, the centrifugal force will cause a bending of the shaft, which then is said to 'whirl.' Resonance occurs when the angular velocity of the disk-i.e., the forcing frequency-coincides with an eigenfrequency of the shaft. These particular angular velocities are termed 'critical.' In particular, the lowest (smallest) one is often termed 'the critical angular velocity. ' The influence of a small amount of fluid trapped inside an elastically mounted, rotating, and whirling cylindrical vessel was investigated theoretically for the first time by Schmidt [6]. This was followed up, theoretically as well as experimentally, by Kollmann [7]. An interesting and surprising result, found first by Schmidt [6] and verified by Kollmann [7], is that, by synchronous whirl-where the whirling frequency of the contained fluid 'ring' is equal to that of the shaft-the critical angular velocity of a rotor partially filled with fluid is identical with that of a rotor completely filled with fluid. In other words, it is insignificant how much fluid the rotor actually contains, as long as the cylinder wall is fully wetted. It is noteworthy that Schmidt [6] found this result not through mathematical analysis but solely through physical reasoning. His argument was that, starting out with a certain amount of fluid and then adding more, the added fluid will distribute itself concentrically about the neutral axis; and the elastic forces will not be changed. Hence, the critical angular velocity will also remain unmodified. The fluid acts as added mass, lowering the critical angular velocity, to a value that sometimes is termed, 'the reduced critical speed. ' Continued research allowed for whirl motion of the rotor and the contained fluid with an angular velocity which is different from that of the rotor itself, in other words, asynchronous whirl. Such motion had been observed by Kollmann and is described in his 1962 paper [7], but the phenomenon remained unexplained. Wolf [8] found, in 1968, that the critical angular velocity by asynchronous whirl always is higher than the critical value for synchronous whirl. Also, he found that by asynchronous whirl, the critical angular velocity is no longer independent of the filling ratio. Kuipers [9] gave a similar analysis, actually some years earlier than Wolf, namely in 1964. Through his analysis, Kuipers [10] was also capable of explaining and verifying the experimental results of Kollmann [7] for asynchronous whirl. While Wolf [8] assumed circular whirl, which can be described by just one rotor degree of freedom, Kuipers [9] formulated his theory for a rotor with two degrees of freedom. However, as shown also by Kuiper himself, when the two stiffness and two damping parameters for the two degrees of freedom system are equal, the system actually reduces to a one degree of freedom system.
Ehrich [11] considered asynchronous as well as synchronous whirl in an analysis that preceded the one of Wolf [8] slightly, but his study of asynchronous whirl was based on a highly simplified fluid model which assumed that the centrifugal force and the viscous shear force were the dominating forces; in other words, pressure gradient, momentum, and Coriolis force terms were neglected. It is noted that Ehrich gave a simplified analysis also for the case of synchronous whirl where the cavity containing fluid is only partially wetted.
A difference in formulations of the works mentioned so far should be pointed out. For their studies of synchronous whirl, Kollmann [7] and Ehrich [11] considered unbalanced rotors, where the center of mass does not coincide with the geometric center. As mentioned a little earlier, this gives a forced vibration problem, or, in terms of stability, a resonance problem. For their studies of asynchronous whirl, Kuipers [9] and Wolf [8] considered perfectly balanced rotors. In this case, 'internal resonance' occurs when the whirl angular velocity of the rotor coincides with the propagation speed of the fluid (shallow water) wave [8,12]. The fluid force is proportional to the acceleration [8]; hence, the governing equation is homogeneous and we have a selfexcited vibration problem. It is thus not a 'resonance problem' as such. Mathematically, the problem takes the form of an eigenvalue problem, and the critical angular velocity corresponds to the onset of a dynamic instability, also called flutter [2,13].
Hendricks and Morton [14] investigated, by asynchronous whirl, the interplay between viscosity of the fluid and damping of the rotor motion, and the effect of these parameters on the critical angular velocity. The former effect (viscous 'fluid damping') was based on a boundary layer model. Holm-Christensen and Träger [15] considered, for the same problem formulation, a more rigorous representation of fluid viscosity by employing directly the linearized Navier-Stokes equations. In this way, they were able to refute some of Hendricks and Morton's [14] findings and to find a curious 'instability top,' implying a lower critical angular velocity than previously thought.
We remark, finally, that a large body of work on 'the elastically mounted rotor with fluid' problem has been carried out also in Japan. A review of this work, of which the majority has been published only in the Japanese language, has been given by Kaneko et al. [16].
A different yet closely related area of research is concerned with the behavior of a shallow fluid layer, or thin-film flow, within-or exterior to-a rigidly mounted rotating cylinder. Phillips [17] analyzed the flow within a rotating horizontal cylinder, in the case where the gravitational force is relatively weak in comparison with the centrifugal force. Miles and Troesch [18] considered the case where the effect of gravity is absent (or ignored) altogether. Further details of the gravity-affected problem (often referred to as a rimming flow), such as the effect of inertia, viscosity, surface tension, and the existence of shock wave solutions, have been investigated by many researchers, e.g., Hosoi and Mahadevan [19], Ashmore et al. [20], Acrivos and Jin [21], Benilov et al. [22], and Badali et al. [23], just to mention a few. With a view to the before mentioned analysis of Ehrich [11], it is noted that many of the studies into rimming flows also consider just partially wetted surfaces. A comprehensive review paper, covering studies up to 2011, has been published by Seiden and Thomas [24].
Pukhnachev 1 [27] and Moffatt [28] investigated the case where a viscous fluid film is on the outside of a rotating cylinder. These papers likewise inspired many researchers to work toward a more detailed understanding of the problem, e.g., Hinch and Kelmanson [29], Pukhnachov [25,26], and Kelmanson [30].
Returning to the elastically mounted rotor problem, some of the initial investigations mentioned above were based on a concern that fluid might be trapped inside a rotor by accident, for example, as a result of condensation [4,11]. But actually, the presence of a small amount of fluid inside a rotor is not all bad; it is known that the fluid, at supercritical angular velocities (i.e., past resonance), can stabilize an unbalanced, whirling rotor, in the sense that it can act as a counterbalance and thus limit the amplitude of the whirling motion [5]. This is a topic that has seen renewed interest in recent years, with applications to passive control of rotating machinery. Hence the name 'fluid balancer' has been coined [16,31]. An earlier paper [32] aimed at giving an explanation of the basic mechanism behind such an application. An unbalanced rotor gives a forced vibration problem and it is normally assumed that this causes a synchronous whirl problem also when the rotor contains fluid [7,11,33]. This situation would be desirable, since 'perfect balancing' could then be achieved. However, experiments [31] have shown that the fluid does not always follow the rotational movement of the rotor and the unbalanced mass (by experiments, typically a small, added weight) synchronously, but experience a drift. This indicates asynchronous whirl of the rotor and the contained fluid; hence this was assumed in [32] in the formulation of the mathematical model. It is noted that later experimental results [34] also corroborate this assumption. Spannan et al. [34] state that, "The balancing effect is not as distinct as expected...," and they state that their results, "... stand in line with [32], who identified asynchronous whirl as the driving counterbalancing phenomenon." It may be noted here, in passing, that unbalanced whirling rotors equipped with a 'ball balancer'-a device which utilizes a number of steel balls (of ball bearing type) rather than a fluid-typically display a similar type of drift/asynchronous whirl [35][36][37][38], in particular in an angular velocity range not far away from a critical (resonance) angular velocity.
The modeling of the fluid layer in [32] followed the shallow water wave approach of Berman et al. [12]. The scaling used in the perturbation analysis was also similar to the one used in that paper, with the ratio between the fluid mass and the mass of the empty rotor playing the role of the basic small parameter in the problem. However, in some applications, this parameter is not necessarily so small. A more recent paper [39] made use of a more appropriate scaling, assuming that the basic small parameter is of the order (fluid layer thickness)/(vessel radius). This scaling is employed also in the present paper. It may be noted that the shallow water wave formulation of Berman et al. [12] has been employed also by Colding-Jørgensen [40,41], Kasahara et al. [42], and Yoshizumi [43,44]. In the analysis of Colding-Jørgensen [40], the dispersive term was neglected in the equation governing the fluid layer thickness. This equation then reduces to an algebraic equation, admitting a solution which represents a simplified hydraulic jump. (This solution was studied also by Berman et al. [12].) Kasahara et al. [42] solved the shallow water equations numerically, employing a finite difference approach. These authors found numerical indications of solitary waves, also known as solitons. In the first of two papers by Yoshizumi [43], while the shallow water approximation is employed, all terms that appear under this formulation are included in the analysis, with no further order-of-magnitude considerations. In the second paper [44], order-of-magnitude considerations are employed, leading to a more manageable analysis.
The experiments of Berman et al. [12] indicated the appearance of a soliton. However, the existence of such a wave in a shallow fluid layer in a cylindrical domain-i.e., for a problem with periodic boundary conditions-is not obvious, as it is recalled that the soliton is a limit case of the cnoidal wave, as the period goes to infinity [45]. In workin-progress, we have employed the method of matched asymptotic expansions to the forced Korteweg-de Vries-Burgers (KdVB) equation. In the boundary layer approximation, this equation actually reduces to an unforced Korteweg-de Vries equation, which has a soliton solution. (See also Ref. [46], described in more detail a bit later.) This work will appear in a forthcoming paper by the present authors.
In a different setting, solitary waves have been detected on the free surface of a rotating fluid which completely fills a cylindrical container [47,48], like a drink stirred up in a glass. But the problem of the present paper may be closer related to that of resonant forcing of shallow water in a tank, investigated theoretically by Chester [49] and experimentally by Chester and Bones [50]. Cox and Mortell [51] showed that the problem can be formulated as a forced Korteweg-de Vries equation. This equation was solved numerically. The existence of a solution to a forced Korteweg-de Vries equation was proved by Benjamin et al. [52]. Wu [53] showed how a moving, forced disturbance in shallow water, modeled as a forced Korteweg-de Vries equation, can generate a periodic succession of solitons. The stability of these waves was investigated by Camassa and Wu [54]. Amundsen et al. [46] presented an approximate analytical solution to a forced KdVB equation, based on the method of matched asymptotic expansions. Amundsen et al. [46] list also many other papers which were inspired by the studies of Chester and Bones [49,50]. It is noted that all of these studies fall within the topic of sloshing. Fundamental aspects of this topic are reviewed in Moiseev [55] and Moiseev and Petrov [56]. Cooker [57] considered an interesting extended version of 'the pendulum problem' described in Moiseev [55]. Industrial sloshing problems, and sloshing models typically employed in industry, are reviewed in Kaneko et al. [16].
The problem formulation of the present paper is similar to that of the earlier ones [32,39]. However, contrary to the Lindstedt-Poincaré perturbation method employed in [32,39], this paper gives a more systematic approach by making use of the method of multiple scales from the outset. By either method, the variation in the fluid layer thickness is found to be governed by a forced KdVB equation [58]. But the multiple scales formulation resolves a number of delicate scaling issues that have not been treated satisfactorily in the earlier papers (including Berman et al. [12]), such as the presence of a slow-time derivative in the forced KdVB equation. From a physical point of view, the main aim of the paper is to give an analytical explanation of how the fluid contained in the rotor can counteract the effect of the unbalanced mass, in other words, to provide analytical understanding of the mechanics of the fluid balancer. The main interest is in the angular velocity range around (i.e., not far away from) resonance. Based on the experimental results mentioned earlier [31,[34][35][36][37][38], a condition of asynchronous whirl of the fluid is assumed. That is to say, the forcing due to the unbalanced mass is not assumed to be 'hard' enough to synchronize the fluid motion to the forcing generated by the unbalanced mass. This gives an inherent imperfect balancing effect, where the fluid slowly but unavoidably will drift away from its optimum position; but again, this is in line with the experiments.
The paper is organized as follows. The rotor equations are stated in Sect. 2, and the fluid equations in Sect. 3. The presence of a point mass (as a model of unbalanced mass) necessitates the formulation of a rotor model with two degrees of freedom (two rotor displacements). This, in turn, necessitates also the presence of two forcing terms in the KdVB equation. However, as will be shown later (in Sect. 7), when the damping and stiffness parameters are equal for the two degrees of freedom, the two rotor displacements, i.e., the two degrees of freedom, can be combined into one complex displacement. This approach is in contrast to that of Berman et al. [12] and Colding-Jørgensen [40,41] who, for a perfectly balanced rotor, employed a one degree of freedom rotor model at the outset. Nondimensionalization of the governing equations is discussed in Sect. 4. A multiple scales solution of the fluid equations is the subject of Sect. 5. Under the assumption of asynchronous whirl, the analysis gives, to the leading order, a frequency equation which, for a given angular velocity, determines the whirl frequency. The next order gives, as also mentioned above, a nonlinear, forced, KdVB equation. A perturbation approach to this equation, yet again based on the method of multiple scales, is discussed in Sect. 6. We consider the regular perturbation problem, where the dispersive term is considered non-small, as well as the more interesting, and more physically correct, singular perturbation problem, where the dispersive term is small. The coupled fluid-structure system is the subject of Sect. 7. The governing equation of the fluid-loaded rotor is a forced Mathieu-Hill type equation, where the forcing term represents the unbalanced mass and a time-dependent stiffness term represents a 'drift' of the fluid relative to the unbalanced mass. Seen from a fixed coordinate system, the said drift manifests itself as 'beats.' This is due to interaction between the whirl frequency and rotation frequency, which are non-equal but close. Section 7 includes also comparison with experiments, as well as a discussion on the basic mechanism of the fluid balancer. Finally, the main results of the paper are summarized, and conclusions are made, in Sect. 8.

The rotor equations of motion
Consider a rotor, in the form of a rotating vessel/fluid chamber of mass M, equipped with a small unbalanced mass m located a distance s from the geometric center, and containing a small amount of fluid, spun out to form a thin fluid layer on the inner surface of the vessel, as sketched in Fig. 1. The angular velocity of the rotor is Ω, and it is assumed to be in a state of asynchronous whirl, with an angular velocity ω which is (slightly) different from Ω. The inner radius of the vessel is R. In terms of a coordinate system (x, y) attached to the rotor wall, the thickness of the fluid layer at time t and position x is given by h(t, x), 0 ≤ x ≤ 2π R. The rotor is supported by springs with spring constants K x and K y , in theX andȲ directions, respectively, of a space-fixed coordinate system (X ,Ȳ ). The structural damping forces in both of these directions are proportional to the parameter C. The rotor thus has two degrees of freedom. In terms of this (fixed) coordinate system, the matrix equation of motion Fig. 1 Sketch of the basic configuration, with definition of the space-fixed coordinate system (X ,Ȳ ), the rotating (rotor-fixed) coordinate system (x,ȳ), and the rotor-wall-fixed coordinate system (x, y). The mass of the empty rotor is M, while the unbalanced mass is m; this mass is placed a distance s from the center. K x and K y are the spring constants in theX and theȲ direction, respectively, while C denotes the damping parameter in both of these directions. The angular velocity of the rotor is Ω, while ω is the angular velocity of the whirling motion of the rotor. Finally, the thickness of the fluid layer at time t and position x is given by h(t, x) of the rotor is given by Here, an 'overdot' denotes differentiation with respect to time t. X r and Y r are the deflections of the rotor, F Xr , F Y r are the fluid force components acting on the rotor, and as mentioned a little earlier, Ω is the angular velocity of the rotor and ω is the angular velocity of the whirling motion of the rotor. As indicated by the last term on the right-hand side of (1), the fluid forces depend on both of these frequencies. (This will be made clear in Sects. 5, 6, and 7.) The first term on the right-hand side shows that, in a fixed coordinate system, the unbalanced mass introduces a periodic forcing. The matrix equation of motion in terms of a coordinate system (x,ȳ) fixed to the rotor is where x r and y r are the rotor deflections in thex andȳ direction, respectively. Here, the first term on the right-hand side shows that, in the rotating coordinate system, the unbalanced mass introduces a time-independent force proportional to Ω 2 , acting in thex-direction.

General equations
The Navier-Stokes equation in a rotating coordinate system is given by [59, p. 147], where u = {u v} T is the velocity vector, D/Dt denotes the material derivative (i.e., D/Dt = ∂/∂t + u · ∇), p is the pressure, ρ is the density, and μ is the dynamic viscosity of the fluid. The fictional body force F, acting in a rotating reference frame, is given by ( [59], p. 140) Here −f 0 = {F G} T is an apparent body force that compensates for the translational acceleration of the frame, is the angular velocity vector, and r is the radius vector from the origin of the rotor-fixed coordinate system (x,ȳ) to the cylinder wall. Thus, −2 ×u is the Coriolis force, and − ×( ×r) is the centrifugal force. The continuity equation is

The shallow water equations
The fluid motion in the rotating vessel will be described by a shallow water wave approximation of the Navier-Stokes equations, and in terms of a coordinate system (x, y) attached to the wall of the rotor, as shown in Fig. 1. x and y are rectangular (Cartesian) coordinates, indicating that curvature effects will be ignored. This is permissible when the fluid layer thickness h(t, x) is sufficiently small in comparison with the vessel radius R, i.e., |h(t, x)|/R 1 for all x, t. Ignoring gravitational forces also, the fluid equations of motion can be written as [12,60] Here u and v are the fluid velocity components in the x and y directions, ν = μ/ρ is the kinematic viscosity of the fluid, and p and ρ are again the fluid pressure and fluid density, respectively. The body force F =Ẍ r sin (x/R) −Ÿ r cos (x/R) is given relative to the rotor-fixed coordinate system (x,ȳ), where the acceleration vector {Ẍ rŸr } T is given by Strictly speaking, a body force term in the form is present on the right-hand side of (7), as well as a viscous term in the form ν∂ 2 v/∂ y 2 . These terms have been dropped here, since the fluid layer is thin (in comparison with R). The simplification of (3) and (4) to the form (6), (7) is discussed in the "Appendix." The continuity equation (5) takes the form The boundary conditions are where again, h(t, x) specifies the free surface of the fluid layer.
In the shallow water approximation, it is assumed that where h 0 is the mean fluid depth. Using this relation, (7) can be written as This equation can be integrated with respect to y, to give Differentiating this equation with respect to x and inserting it into (6), we get Let denote the mean flow velocity in the x-direction. Applying this 'operator' to (15), we get where the following models (i.e., approximations) have been employed [12,62]: These equations are models for dissipation due to wall friction and internal fluid friction, respectively. In the first equation, β is a friction coefficient ( [62], p. 86; [63], p. 682), known, e.g., from head loss in pipe flow (e.g., [64], Chap. III); and ν ev in the second equation is the eddy viscosity coefficient of Boussinesq ([60], p. 484; [62], p. 536). Applying (16) to the continuity equation (10), the latter can be written as

Nondimensionalization
We will now write h(t, where h 0 is again the mean fluid depth and h is the height perturbation therefrom. Next, in order to recast the governing equations into nondimensional form, we introduce the parameters , In terms of standard shallow water wave theory [60], the parameter c 0 corresponds to the shallow water wave speed (gh 0 ) 1 2 , but here the gravity acceleration g is replaced by the centrifugal acceleration RΩ 2 .

The fluid equations
A nondimensional version of (17) can now be obtained as Similarly, the continuity equation (19) can be written as

The rotor matrix equation
Applying (20) to (2), we obtain, for a rotating coordinate system, the nondimensional form where a dash denotes differentiation with respect to the nondimensional time t * .
In terms of the fixed coordinate system, we get the nondimensional form

Multiple scales solution of the fluid equations
In order to find approximate, analytical solutions to (21), (23), we will employ the method of multiple scales [65,66]. Let We have then Expansion of (21) and (23) gives In order to solve (28), the 'traveling wave' variable is introduced. Here ω * is the nondimensional whirling frequency [defined in (20)]. Inserting (30) into (28), we obtain These equations only have non-trivial solutions if which gives The traveling wave definition (30) gives that these frequencies correspond to the possible wave speeds c ± = δ ± (1 + δ 2 ) 1 2 . The '+ solution' corresponds to a progressive (forward traveling) wave and the '− solution' to a retrograde (backward traveling) wave. Experiments show that only the latter type exists [12], that is, is stable; accordingly, the '− solution' is used in the following. This means that the whirling frequency ω * is slightly lower than the rotor frequency Ω * , i.e., that ω * /Ω * = 1 + δc − < 1. The continuity equation (31b) then gives Consider the body force F, that is, the terms F =Ẍ * sin x * −Ÿ * cos x * in (29a), as well as the (ignored) body force G =Ẍ * cos x * +Ÿ * sin x * defined by (9). It is noted that we can write and similarly for the cos x * term. Employing these expansions, we can write In order to transform the rotor-fixed body force components {F, G} T into the traveling wave-fixed components 2 {F w , G w } T , we need the transformation 3 which gives the simple form Employing the first of these expressions, as well as (34), the terms of order δ 2 in the expansion of (21) (that is, (29a)) can now be written where It is noted that Having introduced the traveling wave variable (30), which depends on the fast time t 0 , we will assume that the rotor motions depend on the slow time scale t 1 only and not on the fast time scale t 0 . Thus,Ẍ * = δ 2 ∂ 2 x * r ∂t 2 Equation (39) is a forced Korteweg-de Vries-Burgers (KdVB) equation [58]. Without dissipation (D 1 = E 1 = 0) and external forcing, it reduces to the Korteweg-de Vries equation. The Burgers equation is obtained with C 1 = E 1 = 0 and again no external forcing. In the following, we will find approximate analytical solutions to (39). These solutions must satisfy the conditions of periodicity, as well as the condition of conservation of mass, π −π h 1 (ξ )dξ = 0. (43) 6 Multiple scales solution of the forced KdVB equation

A regular perturbation problem for C 1 non-small
It is our purpose to obtain approximate, analytical solutions to the coupled fluidstructure system. While we have employed the method of multiple scales, we found that the O(δ 2 ) equation (29a)-which was developed into (39)-turned out to be nonlinear. Since we aim at an approximate, analytical solution, we will reconsider the magnitude of the coefficients of (39) and employ a perturbation method also to this equation. In the following, we will consider only the case E 1 = 0. Modifying the traveling wave variable (30) to the form where c 1 is a constant, we have the relation Equation (39) can then be written where we have introduced the parameter Υ = −c 1 A 1 . Now (46) is written in the form where Here is a 'bookkeeping parameter' that is used to emphasize the smallness of the proceeding coefficients [66]. After the solution of the problem, is again set equal to one. Integrating once, we obtain where K is an integration constant. It is noted that λ plays the role of an eigenvalue. Let We have then We will also writê whereẑ 1 =x 1 + iŷ 1 ,ẑ * 1 =x 1 − iŷ 1 , i.e., the complex conjugate, and cc stands for the complex conjugate of the preceding term. Inserting (50) and (51) into (49), and ordering the terms, we obtain where we have introduced K = K 0 + K 1 +· · · Setting K 0 = 0, we find the complete solution to (53) as In order to eliminate secular terms in this solution, (54) gives the solvability condition Introducing the complex representation A(X 1 ) = 1 2 α(X 1 )e iβ(X 1 ) , we obtain the solution where α 0 and β 0 are constants. It is, however, clear at this point that we don't need to include λ 1 , that is to say, we can set λ 1 = 0, since H 0 is periodic as long as A is 'sufficiently constant. ' We then obtain, finally, the uniformly valid solution It is noted that O( ) here indicates O(d 1 ). Restoring the original variables, and expressing the first term in a different form, we have where a 0 and b 0 are constants. The second, 'particular solution,' term of (59) satisfies the periodicity conditions (42) 'automatically.' Inserting the first term of (59) into (42) gives, for sufficiently small d 1 = D 1 /C 1 , the condition sin λ 1/2 0 π = 0. Thus, λ 1/2 0 = n, or λ 0 = n 2 , n = 1, 2, . . . However, the case n = 1 corresponds to primary resonance, which must be investigated specifically; this will be done in the following (below). The solution (59) is thus valid only for n = 2, 3, . . . It is noted that (59), again for sufficiently small d 1 = D 1 /C 1 , satisfies the conservation of mass condition (43). Finally, it is noted that in terms of the 'original' variable Υ , we have Υ = Υ n = n 2 C 1 , n = 2, 3, . . .
When λ ≈ 1, we have to scale (order) the variables as follows: and in the following we will set λ = 1 + υ 1 + · · · , where υ 1 is a so-called detuning parameter, used to be able to tune λ away from the resonant condition λ = 1. We have, then, Setting K 0 = 0, we find the complete solution to (61) as In order to eliminate secular terms in this solution, (62) gives the solvability condition Introducing again the complex representation A(X 1 ) = 1 2 α(X 1 )e iβ(X 1 ) , we obtain the solution Inserting into (63), we thus obtain the uniformly valid solution Restoring the original variables, we obtain The periodicity conditions (42) are satisfied, for sufficiently small d 1 = D 1 /C 1 , with β 0 = 0.

A singular perturbation problem for C 1 small
The previous results are not useful for cases where C 1 is small compared to B 1 and D 1 . This is, however, typically so, since δ is small. Here we will consider this case. Specifically, we will reconsider (39) with the following ordering of the terms: where again is a 'bookkeeping' parameter that indicates the smallness of the proceeding terms. Since the highest order derivative in (68) is multiplied by a small parameter, the solution of (68) will be a singular perturbation problem. Integration gives where K is an integration constant, and λ := Υ , to emphasize that this parameter plays the role of an eigenvalue. We will again employ the method of multiple scales and expand h 1 (ξ ) as where θ 0 and θ 1 are given by It is noted that θ 0 is a so-called stretched variable, or a boundary layer variable. Then and let z * r = x * r + iy * r , z * * r = x * r − iy * r . We obtain It is necessary to set K 0 = K 1 = 0 in order to have a bounded solution. Then, we obtain the complete solution to (74) in the form where the functions A and B are to determined in the following. Inserting (76) into (75), we find that, in order to avoid secular terms in the solution of (75), the following solvability conditions need to be satisfied: We obtain where a 0 and b 0 are constants. The complete solution to (74) is thus given by In terms of the original variables, we obtain Also in the present case, the 'particular solution' part of (81) satisfies the periodicity conditions (42). However, the 'homogeneous' part of the solution (81) cannot satisfy periodicity. Accordingly, it is necessary to set a 0 = b 0 = 0. The eigenvalue λ 0 will be determined approximately by employing Rayleigh's principle, whereh 1 is a test function that satisfies the periodicity conditions (42). Choosing h 1 (ξ ) = cos nξ and using (42), we obtain It is noted that these eigenvalues are completely equivalent to the eigenvalues Υ n found in connection with (59).
To sum up the results of this section, using only the 'particular solutions,' we have x * r cos ξ + y * r sin ξ , C 1 non-small, Υ = C 1 , n = 2, 3, . . . , (a) For each of the cases considered, we can go on and evaluate the perturbation expansion to higher order, i.e., evaluate the nonlinear response. Still, the linear approximations can provide understanding of the basic mechanism of the fluid balancer, since the unbalanced mass acts as a forced excitation which will provide a first approximation to the rotor amplitude. The mechanism of the fluid balancer will be discussed at the end of the following section.
7 The coupled fluid-structure system

Fluid forces
The nondimensional version of the pressure equation (14), evaluated on the vessel surface y = 0, takes the form where the last approximation is made on the assumption that δ is small. The fluid force components, acting in the radial and the tangential direction relative to the traveling wave, are given by where Here M fluid is the mass of the contained fluid, with w being the width of the vessel (the height 'out of the paper' in Fig. 1).
The connection between the traveling wave-fixed force components F * r , F * t and the rotor-fixed force components F * xr , F * yr , appearing in (24), is given by Since ω * is close to Ω * , the coefficients in the matrix are slowly varying parameters, and we will write ( [67], p. 11) which is true for non-large times t * (t 0 ). Employing (85), (86), we have: (i) for C 1 non-small, Υ = C 1 : (ii) for C 1 non-small, Υ ≈ C 1 : and (iii) for C 1 small:

Rotor equation of motion
Let z * r = x * r + iy * r , and let F * zr = F * xr + iF * yr = (F * r + iF * t )e iκ * t * = F z * r e iκ * t * , where κ * = ω * − Ω * , and F = F R + iF I . Then, with χ = 1, and recalling that the rotor motions are assumed to depend on the slow time t 1 only [cf. (41)], the system (24) can be written in complex form as with That is to say, with χ = 1, the rotor system reduces to one with just one (complex) degree of freedom. It is noted that . It is noted, also, that C 1 ∝ Ω 2 * , D 1 ∝ Ω * . Since (93) is an equation of Hill's type, a perturbation method approach is yet again necessary. It is noted that (93) is not a singular perturbation problem, since the transformation t * = t 1 /δ converts it into a regular problem. Considering a simple ('pedestrian') perturbation expansion in the form we obtain It will be seen that the solution (95) is stable. Writing F = F R +iF I , the leading-order term is Since dz 0 /dt 1 = 0, the solution of (97) with respect to z 1 is very simple. We find In the present context, we will be interested mainly in the leading-order solution (99), as this term captures the essential 'mechanism' of the fluid balancer. This will be made clear in the following subsection. It is interesting to notice, however, that the second term, z 1 , introduces a modulation of the motion. Seen from a fixed coordinate system, this modulation will occur in the form of 'beats.' This can be seen as follows. From (100), z 1 ∝ 1 − exp i κ * δ t 1 . In terms of a fixed coordinate system [cf. (25)], let the complex displacement Z * r = X * r + iY * r . The relation between Z * r and the displacement z * r in the rotating coordinate system is given by Z * r = z * r exp(iΩ * t * ). Thus, expressed in terms of the fast time t * , with Z * r = Z 0 + δ Z 1 + · · · , Since the angular velocity of the whirl ω * = (1 + δc − )Ω * is just slightly smaller than the angular velocity of the rotation Ω * [cf. (33)], the factor sin 1 2 (ω * − Ω * )t * gives the well-known slowly varying amplitude, while cos 1 2 (ω * + Ω * )t * ≈ cos Ω * t * (and likewise for the similar sin-term). Kollmann [7], and more recently Spannan et al. [34], made experimental observations of beats (beat-type oscillations) of a slightly unbalanced rotor partially filled with fluid. Similar observations were made by Berman et al. [12] in their experiments. Also related to the present system, computations with beat-type oscillations of an unbalanced rotor equipped with a ball balancer are discussed in [38]. Figure 2 shows a numerical evaluation of (95) based on just (99), i.e. for the approximation z * r ≈ z 0 , with the fluid loading parameter F evaluated according to (94 c), with n = 1. The data used are in accordance with experiments of Nakamura [31], also referred to in Ref. [32]. Part (a) shows x * r = Re(z * r ), part (b) shows y * r = Im(z * r ),  respectively. These values of C 1 are certainly small. Assuming that ν * = 0.04, the corresponding values of D 1 are 7.32 × 10 −2 , 6.56 × 10 −2 , and 6.05 × 10 −2 , respectively. The structural damping parameter ζ was chosen as ζ = 0.01. The specific value of this parameter is not very critical. Figure 2(c) shows a reasonably good agreement with the experimental results of [31] for all three filling ratios. Figure 3 shows similar (to Fig. 2) results for the case where the fluid loading parameter F is evaluated according to (94a), with n = 2. All parameter values are as those used in Fig. 2, except for the value of ζ , which was set to ζ = 0.10. The assumed nonsmallness of C 1 in the definition (94a) is clearly violated in this case. The consequence is that the fluid loading parameter is largely overestimated. This is clear directly from the form of F, as given by (94a). However, due to the simplicity of this case, the basic mechanism of the fluid balancer is more readily understood. Considering Fig. 3(a), it is noticed that x * r changes sign, from positive to negative, across the resonance point. Recalling that C 1 > 0, (84 a) then shows that h 1 (ξ ) > 0 around ξ = 0 for subcritical values of Ω * , while h 1 (ξ ) < 0 around ξ = 0 for supercritical values of Ω * . That is to say, by subcritical angular velocities, the majority of the fluid is on the same side as the unbalanced mass (i.e., around ξ = 0), whereas at supercritical angular velocities, the majority of the fluid is present oppositely of the unbalanced mass (i.e., around ξ = π ). This explains the fluid balancer in its essence. However, as discussed in the previous subsection, inclusion of higher-order terms will bring a modulation of the motion into play. This will, in turn, introduce a slow but unavoidable drift of the fluid away from its balancing position.

The mechanism of the fluid balancer
Finally, it is interesting to note that the forms of (84b) and (84c), as well as those of (94b) and (94c), are very similar. In particular, for cases where D 1 dominates over C 1 , as in the examples considered here, the results obtained with these two versions of F will be virtually indistinguishable.

Conclusions
In the present paper, we have analyzed the dynamics of the so-called fluid balancer, i.e., a rotor with an unbalanced mass, containing a small amount of fluid. The main aim has been to obtain a solid, analytically based understanding of the balancing effect the fluid can have. Based on experimental evidence, a state of asynchronous whirl has been assumed. This implies an inherently imperfect balancing device where, once a balanced condition is achieved, there will be a slow drift away from it; but again, this is also what experiments indicate will happen.
The method of multiple scales was applied to the shallow water wave equation which has been employed to describe the fluid motion. It was found that the fluid layer thickness perturbation is described by a forced KdVB equation. Two cases were distinguished: one where the fluid layer is just moderately shallow, corresponding to a non-small value of the parameter C 1 multiplying the dispersive term; and one where the fluid layer is truly shallow, corresponding to a small value of the parameter C 1 . In solving the fluid layer equation by again employing the method of multiple scales, the first problem is a regular perturbation problem, while the second one is a singular perturbation problem. An interesting finding is that the mathematical form of fluid layer perturbation thickness (h 1 ) in the latter case is similar to that of the former case, when the condition of resonant forcing of the fluid layer is satisfied. It is noted that resonance is not realized in the former case (with C 1 small).
The theoretical results (based on the leading-order term) for a shallow fluid layer (C 1 small) were compared with (from the literature/earlier work) available experimental results, and good agreement was found. The theoretical results for the just moderately shallow (C 1 non-small), non-resonant case are not comparable with the available experimental results. Nonetheless, these (theoretical) results provide a clear and simple qualitative explanation of the basic mechanism of the fluid balancer. For the case of the truly shallow fluid layer, the dynamics of the fluid layer is less simple and transparent. The higher-order terms of the solution to the coupled rotor-fluid equation include a time-dependent part, which represents a drift of the fluid, away from its balancing position. It was shown that, seen from a fixed coordinate system, the first of these terms represents beat-type oscillations. This beating is due to interaction between the whirl angular velocity ω and the rotation angular velocity Ω, which are non-equal but close.
It would clearly be of interest to continue the present investigation to higher-order terms and, not the least, to possibly find analytical solutions to the forced KdVB equation (39). Also, it would be of interest to investigate the possible transitions from asynchronous whirl to a lock-in to synchronous whirl.

Conflict of interest
The authors declare no competing interests.

Appendix: The shallow water wave approximation
Evaluating the fluid equations of motion (3, 4, 5) in terms of standard polar coordinates (r , θ), with fluid velocity u r θ = (u r , u θ ), followed by a change of coordinates, to the curvilinear coordinates (x, y) = (Rθ, −r ) (cf. Figure 1) and fluid velocity u = (u, v) = (u θ , −u r ), we obtain the following coordinate-form of these equations: In order to estimate the mutual magnitude of the terms in (102) and (103), we note that the vessel radius R is an appropriate length scale of variations in the x-direction, while the mean fluid depth h 0 is an appropriate length scale of variations in the y-direction. It is noted that the ratio h 0 /R is small. We will thus employ = h 0 /R as a small nondimensional parameter in the following. The flow velocity component u, in the x-direction, is of order of magnitude c * = c 0 Ω * , where the speed parameter c 0 and the nondimensional angular velocity Ω * are defined in (20). The variation in the flow velocity component v, in the y-direction, will then be of order of magnitude c * R/h 0 . Thus, we will scale the variables as follows: Additionally, we will introduce the following nondimensional quantities: It is noted that the scalings (105) and (106) are in line with (roughly equivalent to) the ones employed by Stoker ([68], p. 28) in his derivation of the shallow water theory. It is remarked, also, that these scalings are used only here, in the present appendix, not in the main text. Then (102) and (103) can be written 2 ∂v ∂t + 2ū ∂v ∂x + 2v ∂v ∂ȳ The leading terms of (107) and (108) are then: 2Ωū Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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