# Supercritical Coexistence Behavior of Coupled Oscillating Planar Eccentric Rotor/Autobalancer System

Authors: DaeYi Jung
Journal: Shock and Vibration
 (2018)

---

## Abstract

The automatic balancing and undesirable nonsynchronous behavior of coupled oscillating configured flexible foundation and planar eccentric rotor equipped with a passive autobalancer (AB) system has been thoroughly investigated here. Specifically, it is described that the unified AB/rotor unit is attached to a foundation via a symmetric support and the foundation is also mounted on the spring-damper isolator which allows oscillating only vertically. Therefore, the AB/rotor unit dynamically interacts with the flexible foundation, which is quite analogous to well-known vertically coupled two-spring and two-mass oscillator. Although the single unit AB/rotor system is widely explored in the related AB studies, such coupled arrangement with AB discussed here has not been previously investigated and thus needs to be explored for further application of AB into various vibration isolation problems of other complicated machines/settings. Therefore, solutions for the synchronous stable balanced and the nonsynchronous unstable limit cycle response of AB/rotor/foundation system are obtained via a fixed equilibrium condition and a harmonic like balancing approach. Furthermore, the stability of each response is assessed via a perturbation and Floquet analysis and, for the system parameters and operating speeds, the undesirable coexistence of the wanted stable balanced synchronous response and undesirable nonsynchronous limit cycle has been thoroughly studied. Due to coupled oscillating feature, it is newly found that the multiple limit cycles are encountered in the range of supercritical speeds and more complicated coexistence is attracted into the system, as well as the damping parameters of coupled components (i.e., flexible foundation) influences of the undesirable limit cycle of AB on the particular supercritical speeds. The findings in this paper yield important insights for researchers wishing to utilize automatic balancing devices in more practical rotor systems coupled with additional vibrating mechanical subsystem such as a washing machine or a reciprocating air conditioning compressor with a flexible foundation.


---

## Body

## 1. Introduction

The mechanical systems and industrial settings, including rotor dynamic systems, are mostly subject to imbalance vibration which causes many critical issues in system operations and maintenance. A remarkable way to manage such imbalance vibration problem is employing the so-called automatic balancing device (ABD). This is a passive device equipped with freely moving eccentric masses that are automatically and naturally adjusted to suppress the imbalance of rotor at supercritical operating speeds. The dynamic interaction between balancer balls and the rotor transverse vibration induces this automatic balancing behavior. The primary advantage of ABD is automatically compensating for imbalance changes without requiring the power, a control system, or sensors. Thearle [1] performed an early experimental work of automatic balancing under various imbalances. Further, [2, 3] verified that a complete balancing can be accomplished via ABD equipped with a minimum of two balancer masses. Majewski [4] investigated the effects of runway eccentricity and rolling resistance of ABD-balls for the rotor-balancer system at steady state. Jinnouchi et al. [5] found that the planar rotor-ABD system can achieve excellent balancing performance in the regime of supercritical speeds but generates amplified vibration at the region of subcritical speeds. Lindell [6] applied ABD to a (hand-held) grinding machine to eliminate the effect of imbalance vibration. J Chung [7, 8] evaluated the stability of dynamic balancing condition of ABD for Jeffcott rotor model and Stodola-Green rotor model based on the characteristic equation. Paul C.P. Chao [9] explored the stability of dynamic modeling for the nonplanar optical disk drive spindles installed with an ABD and showed that the action of ABD significantly reduced the residual radial and tilting vibration.Recently, Kim et al. [10] and Raja. and Bhat [11] proposed the various 3-dimensional dynamic behavior of ABD-rotor system including in-plane transverse, nonplanar tilting, and gyroscopic effect and revealed that the supercritical balancing behavior can be achieved. Even though ABD shows the effectiveness of dealing with the supercritical unbalance vibration, [12, 13] investigated and remarked the supercritical coexistence of stable synchronous balancing condition and unstable limit cycle, which induces a significant undesirable vibration.D.J. Rodrigues [14] performed the bifurcation analysis for the in-/out-plane motions of a two-plane ABD-rotor system and noted the undesirable coexistence of a stable balanced state together with other less desirable equilibriums via numerical simulations. J. Ehyaei [15] explored the stable balancing behavior of unbalanced flexible rotating shaft equipped with multiple ABDs. DeSmidt [16] provides all fixed equilibriums of flexible shaft/rigid rotor/ABD system together with solution search processes, and the stability of proposed system about those fixed equilibriums is also investigated. Haidar [17] includes the collision and friction effects of balancing masses into the general model for automatic balancing of supercritical shafts and compared the proposed model predictions with experimental values. Razaee [18] presented the new spring coupled balancer masses based ABB reducing the imbalance vibration amplitude at the speeds below the 1st critical speed and it guaranteed more margin of stable region for the perfect balancing condition. Ozan [19] proposed a new flap based mass distribution control system adjusting the orientations of the flaps and moving the COG of the balancers to an appropriate location to compensate the imbalance vibration. Furthermore, Inoue, T., Ishida [20] also validated the limit cycle analysis of planar ABD-rotor with the experimental results. Furthermore, [21] showed three different types of limit cycle behavior of the planar dual-ball automatic balancer: a pure-rotary periodic motion, a pure oscillatory periodic motion, and a compound-rotary periodic motion. Recently, Jung and DeSmidt [22–24] presented the analytical approach for merged synchronous responses and unstable nonsynchronous supercritical limit cycle of single-plane ABD/rotor system based on harmonic like balancing approach. Although such nonlinear and automatic balancing behavior of ideally configured single unit AB/rotor with symmetric supports (or bearings) has been widely studied and is generally understood, an investigation about the coupled behavior of AB/rotor system with other oscillating subsystems has not been fully conducted. Therefore, in this study, the model of coupled oscillating planar AB/eccentric rotor with flexible foundation structure is considered and the automatic balancing and nonlinear behavior of proposed system has been thoroughly explored. This kind of setting can be possibly applicable to many mechanical systems containing the imbalance natured rotor such as a washing machine or a reciprocating air conditioning compressor equipped with a flexible foundation. Especially, for given system parameters, this work focuses on undesirable coexistence, nonsynchronous limit cycle, and synchronous balanced condition, along the supercritical speeds where the balancing performance of AB can be generally maximized. The contents of the paper are organized as follows: First, the solutions for the synchronous stable balancing and the nonsynchronous limit cycle responses of the system are obtained via a fixed equilibrium analysis and a harmonic like balancing approach. Furthermore, for given system parameters, the stability of each response is assessed via a perturbation and Floquet analysis. Finally, the analysis is numerically investigated.


## 2. Coupled Oscillating Rotor/Autobalancer System with Flexible Foundation

In this section, the equations-of-motion of planar-rigid rotor/dual mass autobalancer (AB) system supported by flexible foundation are derived. Specifically, as depicted in Figure1, the AB/rotor unit is mounted on a symmetric flexible support, and the foundation coupled with the AB/rotor unit is supported by spring-damper isolator connected to the ground and allowed to vibrate only vertically. Consequently, the AB/rotor unit dynamically interacts with the flexible foundation; thus this system is quite in accordance with two-spring and two-mass coupled oscillator in the vertical direction. Compared to the previous works [20–23] focusing on a single unit of AB-rotor-bearing system, the eccentricity of this study is that the AB-rotor-bearing unit has been connected with an additional component, flexible foundation, in coupled oscillating manner. Here, the two fixed inertial frames, {N1,N2,N3} with the origin O1 and {n1,n2,n3} with the origin O2, are devised to describe the proposed system. The first fixed frame {N1,N2,N3} is for the movable foundation while the second fixed frame {n1,n2,n3} is for ABD-rotor system. These two frames are vertically aligned as shown in Figure 1. Also, the rotor is assumed to be driven with constant angular speed Ω. Therefore, the transformation from the 2nd fixed inertial frame {n1,n2,n3} to the rotating rotor-fixed frame {a1,a2,a3} is (1)a1a2a3=1000cos⁡Ωtsin⁡Ωt0-sin⁡Ωtcos⁡Ωtn1n2n3.The position vector of the foundation center, F, relative to the 1st fixed frame {N1,N2,N3} is(2)RO1F=yftN3,where yf(t) is the vertical displacement of foundation resulting from foundation support flexibility.Figure 1
Coupled oscillating rotor/ABD system with flexible foundation and the corresponding degree-of-freedom coordinates.The position vector of the unified AB/rotor center referred to as{n1,n2,n3} is also given by(3)RO2C=xtn2+ytn3,where x(t) and y(t) indicate the displacements of AB/rotor caused by the symmetric flexible support connected from the AB/rotor to the foundation.Furthermore, the position vectors of the AB masses are described by(4)Rball.i=RO2C+rbcos⁡ϕita2+sin⁡ϕita3fori=1,2,where ϕ1(t) and ϕ2(t) are balancer mass angular positions measured relative to {a1,a2,a3} and rb is the AB ball track radius.Finally the position of the rotor imbalance is(5)Rim=RO2C+rimcos⁡ϕima2+sin⁡ϕima3,where rim is the imbalance radius and ϕim is the imbalance phase referred to as {a1,a2,a3}.Next, the total system kinetic energy is(6a)T=Tf+Tr+Tim+Tball(6b)Tf+Tr=mf2R˙O1F·R˙O1F+mr2R˙O2C·R˙O2C+Jr2Ω2(6c)Tim+Tball=mim2R˙im·R˙im+mb2∑i=12R˙ball.i·R˙ball.i,where mf, mr, mb, and mim are the mass of a foundation, a rotor, a balancer ball, and an imbalance, respectively. And Jr is the polar moment of rotor.Also, the dissipation energy of system due to relative ball/track motion within the AB and dampings of both rotor and foundation becomes(7)D=cbrb22∑i=12ϕit2+Cr2x˙t2+Cr2y˙t-y˙ft2+Cf2y˙ft2,where cb is a constant viscous damping parameter due to AB ball/track friction and drag effects and Cf and Cr are the damping coefficients of foundation and rotor support, respectively.The strain energy of system due to the springs of both rotor and foundation is given by(8)V=Kr2xt2+Kr2yt-yft2+Kf2yft2,where Kf and Kr are, respectively, the stiffness coefficients of foundation and rotor supports.Using (6a), (6b), and (6c)-(8), the system equations-of-motion are obtained via Lagrange’s equation with corresponding degree-of-freedom vector q≡xtytyftϕ1tϕ2tT∈R5,(9)ddt∂T∂q˙-∂T∂q+∂D∂q˙+∂V∂q=0.The resulting nonlinear equations-of-motion of the foundation/rotor/autobalancer/support system become (10)Mϕ1,ϕ2,tq¨+Cϕ1,ϕ2,tq˙+Kq=Fimt+Fballϕ1,ϕ2,ϕ˙1,ϕ˙2,t,where M(ϕ1,ϕ2,t)∈R5×5, C(ϕ1,ϕ2,ϕ˙1,ϕ˙2,t)∈R5×5, and K∈R5×5 are the system inertia, damping, and stiffness matrices and Fball(ϕ1,ϕ2,ϕ˙1,ϕ˙2,t)∈R5 and Fim(t)∈R5 are the force excitation terms due to the AB and imbalance, respectively. Also, (10) implies the nonautonomous system due to the explicit time variable t. The details of matrices are provided in the appendix.


## 3. Synchronous Balanced Response

In autobalancer application, it is hoped that the system will reach the so-called fixed balanced equilibrium where the balancer balls are synchronized with the rotating rotor and cancel the imbalance effect leading to the zero residual steady-state vibration [16].Using the subnotationss, this balanced equilibrium of (10) is given by (11a)limt→∞q¨t=0limt→∞q˙t=0limt→∞qt=qss=xssyssyf.ssϕ1.ssϕ2.ssTwith the steady-state angular positions of balls (11b)ϕ1.ss=ϕim+cos-1-mimrim2mbrb,ϕ2.ss=ϕim-cos-1-mimrim2mbrb.Here, the condition mimrim≤2mbrb should be necessary for the feasible solution. In other words, the maximum force, mimrimΩ2,due to imbalance should not exceed the maximum AB balancing force, 2mbrbΩ2. Also, (11b) implies that the positions of balls are symmetrically allocated relative to imbalance angular phase where the centrifugal force due to balls is exactly identical to imbalance force. Here, as discussed in [17], due to the friction effect between a balancer mass and a racing track, this ideal and desirable behavior in (11a) and (11b) may not be achieved in reality. In particular, the high friction effect causes significant deviation between the ideal balancing prediction and the actual position of balancer masses. It should be noted that the friction effect should be accounted to predict the actual performance of ABD but is out of this research scope.And the trivial steady-state response of rotor and foundation for balanced equilibrium is(11c)xss=0,yss=0,yf.ss=0.Consequently, it is deduced from (10) that the solution in (11a), (11b), and (11c) satisfies (12)Kqss=Fimt+Fballϕ1.ss,ϕ2.ss,t.Nonsynchronous limit cycle response will be shortly introduced in Section 4; this balanced response in (11a), (11b), and (11c) will be explored in Section 6.


## 4. Nonsynchronous Limit Cycle Response

In this section, the analytical steady-state solution for the limit cycle condition of AB/rotor/foundation system has been obtained.As long as the two balls of AB are fitted in an identical racing track of AB, it is well known from [13, 20–23] and extensive numerical investigations that the two balls of AB are merged together, continuously rotated around the rotor in nonsynchronous manner, resulting in a large whirling of system. And the whirling speed of merged balls is approximated as a constant frequency such as ω∈R. In other words, ϕ(t)=ϕ1(t)=ϕ2(t)=ω·t, where t is a time.Due to merged condition, the system dynamics (10) can be transformed into(13)Mϕ,tq¨+Cϕ,ϕ˙,tq˙+Kq=Fimt+Fballϕ,ϕ˙,twith the reduced coordinates q=xtytyftϕtT∈R4.Next, analytical solution for (13), which is a nonlinear nonautonomous system, is proposed. It is assumed to be a steady-state harmonic solution of rotor and foundation structure:(14a)xsst=A1cos⁡ω+Ωt+A2sin⁡ω+Ωt+A3cos⁡Ωt+A4sin⁡Ωt+x0(14b)ysst=B1cos⁡ω+Ωt+B2sin⁡ω+Ωt+B3cos⁡Ωt+B4sin⁡Ωt+y0(14c)yf.sst=C1cos⁡ω+Ωt+C2sin⁡ω+Ωt+C3cos⁡Ωt+C4sin⁡Ωt+yf.0,where the coefficients, A1, A2, B1, B2, C1, C2, represent the unknown nonsynchronous whirl amplitudes of rotor and foundation and ω is the whirl speed of a merged ball rotating around the AB/rotor. Also, x0, y0, and yf.0 are unknown constant offsets and the coefficients A3, A4, B3, B4, C3, C4 describe the unknown synchronous amplitudes of system. Here, it should be mentioned that the synchronous portion of the response in (14a), (14b), and (14c) is due to rotor imbalance and the nonsynchronous part is due to whirling of the autobalancer masses.Substituting (14a), (14b), and (14c) into (13) yields(i) equation-of-motion aboutx(t)(15a)f11Xcos⁡ω+Ωt+f12Xsin⁡ω+Ωt+g11Ycos⁡Ωt+g12Ysin⁡Ωt+Krx0=0(ii) equation-of-motion abouty(t)(15b)f21Xcos⁡ω+Ωt+f22Xsin⁡ω+Ωt+g21Ycos⁡Ω·t+g22Ysin⁡Ω·t+Kry0-yf.0=0(iii) equation-of-motion aboutyf(t)(15c)f31Xcos⁡ω+Ωt+f32Xsin⁡ω+Ωt+g31Ycos⁡Ω·t+g32Ysin⁡Ω·t+Kryf.0-y0+Kfyf.0=0(iv) equation-of-motion aboutϕ(t)(15d)2cbrbω+A2-B1mbω+Ω2+f41Ycos⁡ω·t+f42Ysin⁡ω·t+f43Ycos⁡ω+2Ωt+f44Ysin⁡ω+2Ωt+f45Xcos⁡2ω+Ωt+f46Xsin⁡2ω+Ωt=0,where unknown vectors X and Y in (15a), (15b), (15c), and (15d), respectively, correspond to A1A2B1B2C1C2ωT and A3A4B3B4C3C4T. The details of coefficients fij(·) and gij(·) in (15a), (15b), (15c), and (15d) are presented in the appendix.By investigating the coefficient of harmonics in (15a), (15b), (15c), and (15d), the following can be addressed:(1)
Based on (15a), (15b), and (15c), it is apparently found that x0=y0=yf.0=0.(2)
Using the harmonic like balancing approach, the synchronous responseY can be obtained from the coefficients of cos⁡[Ω·t] and sin⁡[Ω·t] in (15a) through (15c), which is independent of nonsynchronous response X.(16)g11Y=0,g12Y,g21Y=0,g22Y,g31Y=0,g32Y=0.(3)
The coefficientsf4i(Y)  for  i=1,2,3,4, of cos⁡[ω·t], sin⁡[ω·t], cos⁡[(ω+2Ω)·t], and sin⁡[(ω+2Ω)·t], in (15d) are primarily the function of synchronous response Y, which can be negligible for the limit cycle solution [20, 21]. Also, although the coefficients f4i(X) for i=5,6, are described as a function of X, these are insignificant. This is because the limit cycle response appears with the first harmonic ω+Ω [20, 21].Considering the conditions(1)–(3) above, the following set of nonlinear algebraic equations for the nonsynchronous response X can be extracted from (15a), (15b), (15c), and (15d):(17a)f11X=0,f12X=0,f21X=0,f22X=0,f31X=0,f32X=0.For the simplicity, (17a) can be placed into the following matrix form:(17b)SN·X+EN=0.The details of matrices, SN and EN, are listed in the appendix.Additionally, due to condition(3) above and the assumption that ω is constant, neglecting all harmonics in (15d), the nonlinear equation about the whirl speed ω can be achieved: (18)2cbrbω+A2-B1mbω+Ω2=0.Fundamentally, (18) represents the so-called characteristic equation about the whirl speed, ω, since (18) can be analytically expressed in terms of the polynomials about ω by eliminating A2 and B1 in (18) via (17a) and (17b). In this study, the process to derive the analytical characteristic equation is not presented due to its tediousness, but it should be mentioned that it is doable via X=-SN-1·EN which is another form of (17b). Based on the set of nonlinear equations in (17a), (17b), and (18), the nonsynchronous limit cycle response X∈R7 can be computed. Here, the numerical arc length continuation method [25] is employed to solve for the unknowns X. Applying the harmonic solutions (14a), (14b), and (14c) and reasonable assumptions into (15a), (15b), (15c), and (15d), we can extract the primary steady-state nonsynchronous limit cycle responses only: (19a)xL.sst=A1cosω+Ωt+A2sinω+Ωt(19b)yL.sst=B1cosω+Ωt+B2sinω+Ωt(19c)yf.L.sst=C1cosω+Ωt+C2sinω+Ωt.Here, the synchronous portion has been dropped out due to its insignificant effect under the limiting cycle condition (i.e., ω≠0). In [17, 18, 22], the synchronous portion is also disregarded. In essence, the nonsynchronous vibration amplitudes in (19a), (19b), and (19c) are far superior to the synchronous ones for unstable limit cycles (i.e.,A12+A22≫A32+A42, B12+B22≫B32+B42, C12+C22≫C32+C42) along the supercritical speeds. The proposed solution in (19a), (19b), and (19c) will be numerically discussed in Section 6.


## 5. Stability Analysis

In this section, the local stability analysis about the balanced and limit cycle solutions obtained in Sections3 and 4 has been discussed. To perform this, using perturbation approach, the linearization of original nonlinear equations-of-motion in both (10) and (13) has been carried out and then, based on those linearized equations, the stability of each response is determined by Floquet stability analysis [25].( 1 ) First, to investigate the local stability of the system in the neighborhood of the fixed balanced equilibrium xssyssyf.ssϕ1.ssϕ2.ssT∈R5 in (11a), (11b), and (11c), the nonlinear system dynamics (10) is linearized using the perturbation approach. By considering the small perturbations, Δx(t), Δy(t), Δyf(t), Δϕ1(t), and Δϕ2(t), the perturbed system configurations are set by (20)xt=xss+Δxt,yt=yss+Δyt,yft=yf.ss+Δyft,ϕ1t=ϕ1.ss+Δϕ1t,ϕ2t=ϕ2.ss+Δϕ2t.Now, let us denote a vector of perturbed coordinate as Δq=ΔxtΔytΔyftΔϕ1tΔϕ2tT∈R5 and a vector of fixed balanced equilibrium as qss=xssyssyf.ssϕ1.ssϕ2.ssT∈R5.Substituting (20) into the dynamics (10) and truncating the higher order terms result in a set of linearized equations about the set of steady-state response qss, and the set of linearized equations is given by(21)ΔMqss,tΔq¨+ΔCqss,tΔq˙+ΔKqss,tΔq=0with time-varying matrices ΔM(qss,t)∈R5×5, ΔC(qss,t)∈R5×5, and ΔK(qss,t)∈R5×5 describing the perturbed system about equilibriums.( 2 ) To explore the local stability of the system in the neighborhood of the limit cycle solutions xL.sstyL.sstyf.L.sstω·tT∈R4 in (19a), (19b), and (19c), the nonlinear system dynamics in (13) has been linearized. Similarly, by considering the small perturbations Δx(t), Δy(t), Δyf(t), and Δϕ(t), the perturbed system configurations are set by (22)xt=xL.sst+Δxt,yt=yL.sst+Δyt,yft=yf.L.sst+Δyft,ϕt=ω·t+Δϕt.Based on (22), the resulting linearized equation for (13) is(23)ΔMqL.ss,tΔq¨+ΔCqL.ss,tΔq˙+ΔKqL.ss,tΔq=0with time-varying matrices ΔM(qL.ss,t)∈R4×4, ΔC(qL.ss,t)∈R4×4, and ΔK(qL.ss,t)∈R4×4 describing the perturbed system. Also, Δq and qL.ss in (23) represent a vector of perturbed coordinate ΔxtΔytΔyftΔϕtT∈R4 and a vector of steady-state limit cycle response xL.ssyL.ssyf.L.ssω·t∈R4. Once the linearized equations (21) and (23), which are LTV (linear time-varying) system, are achieved and transformed into the first order form, the stability of both balanced response and limit cycle can be determined using Floquet analysis [25]. Here, the periods of Floquet analysis are Tp=2π/Ω  for (21) and Tp=2π/ω+Ω  for (23), respectively.


## 6. Numerical Simulation Results

In this section, the proposed solutions and approaches in Section3 through Section 5 have been numerically investigated. Specifically, the following are sequentially addressed:( 1 ) The balanced response of system in Section 3 is explored via the time domain simulation of the original nonlinear dynamics in (10) which is constructed in MATLAB/Simulink environment.( 2 ) The steady-state limit cycle solution proposed in Section 4 is compared with the outcome of the numerical simulation of (10).( 3 ) Based on the Floquet analysis in Section 5, at given system parameters and rotor rotational speeds, the stability about the solutions in Sections 3 and 4 has been assessed and then it shows all possible stable regions about solutions.The parameters of system for the numerical simulations are summarized in Table1 and these are used in all subsequent calculations and results unless stated otherwise.Table 1
System parameters for the numerical simulations.
Parameters Masses of rotor and foundation:mr=5 kg/mf= 10 kg Autobalancer parameters(mb,rb,cb): various Imbalance parameters:mim various but mim<mb, rim=rb=0.3 m, ϕim=45∘ Stiffness coefficients of rotor to foundation:Kr=1000 N/m Stiffness coefficients of foundation to ground:Kf=700 N/m (default) or various Damping coefficients of rotor to foundation:Cr=0.001kr (default) or various Damping coefficients of foundation to ground:Cf=0.001kf (default) or variousHere, it should be also mentioned that the given system in (10) possesses multiple critical speeds due to the two-mass coupled oscillating configuration. Considering two conditions, mb≪mr+mf and no stiffness associated with mb, the critical speeds of system are almost identical to the natural frequencies of nominal system (without AB) and given by (24)Ωcr.1=2A-BmfmrΩcr.2=KrmrΩcr.3=2A+Bmfmr,where A=[Kr(mf+mr)+Kymr] and B=-4KrKymfmr+A2. As shown in (24), the natural frequencies are the function of vertically coupled two masses (i.e., AB/rotor unit and foundation) and the associated stiffness. Fundamentally, Ωcr.2 represent the natural frequency associated with the mode of rotor in horizontal direction. On the contrary, both Ωcr.1 and Ωcr.3 are the frequencies related to the vertically coupled modes between AB/rotor unit and foundation. Furthermore, based on given parameters on Table 1, Figure 2 describes the variation of these three natural frequencies as function of Kf and it shows that Ωcr.1 approaches Ωcr.2 as Kf increases and both Ωcr.1 and Ωcr.3 are proportional to Kf. Also, it is found that the lowest natural frequency among Ωcr.1, Ωcr.2, and Ωcr.3 in (24) is apparently Ωcr.1 with the relation Ωcr.1<Ωcr.2<Ωcr.3.Figure 2
Natural frequencies of nominal system (without ABD).With reference to the lowest valueΩcr.1, the nondimensional rotor operating speed and natural frequencies can be obtained via Ω-=Ω/Ωcr.1, Ω-cr.1=Ωcr.1/Ωcr.1, Ω-cr.2=Ωcr.2/Ωcr.2, and Ω-cr.3=Ωcr.3/Ωcr.3 and will be used throughout the paper for clear understanding and observation.


### 6.1. Balanced Response

With the default valueKf=700 N/m, Figure 3 shows the balanced responses on the time domain obtained via (10) based upon the initial conditions of two balls ϕ1(0)=0∘ and ϕ2(0)=180∘ at the three different speeds Ω-=1.25, Ω-=2.58, and Ω-=4.12. Specifically, the results in (a-1) through (a-3) of Figure 3 are obtained at the operating speed Ω-=1.25. And the outcomes in (b-1) through (b-3) are simulated at Ω-=2.58 while the findings in (c-1) through (c-3) are captured at Ω-=4.12. Beside balanced responses, the responses of nominal system (without AB) are presented in Figure 3 and compared with the balancing performances.Figure 3
Balancing responses of system on time domain ((a-1) through (a-3) atΩ-=1.25, (b-1) through (b-3) at Ω-=2.58, and (c-1) through (c-3) at Ω-=4.12).Based onKf=700 N/m and other parameter values in Table 1, we can find that the nondimensional critical speeds are Ω-cr.1=1, Ω-cr.2=2.17, and Ω-cr.3=2.75. Therefore, it is found that (a-1) through (a-3), (b-1) through (b-3), and (c-1) through (c-3), respectively, indicate the balancing responses emerging on the 1st supercritical speeds (i.e., Ω-cr.1=1<Ω-<Ω-cr.2=2.17), the 2nd supercritical speeds (i.e., Ω-cr.2=2.17<Ω-<Ω-cr.3=2.75), and the 3rd supercritical speeds (i.e., Ω->Ω-cr.3=2.75), which are well matched with the outcomes in [16] that the balancing responses are found in each supercritical speed.Moreover, it is observed that the final angular phases of two balls shown in (a-3), (b-3), and (c-3) are identical to the steady-state angular positions of balls, analytically computed based on (11b). The AB parameters used here are mb=0.002 kg and rb=0.3. And the imbalance parameters are mim=0.002 kg and rim=0.3 with ϕim=45∘. Consequently, for certain operating conditions, we can see that the action of AB achieves the perfect balancing of eccentric rotor resulting in zero vertical deflection of foundation as well. Here, it is so obvious that the vertical deflection of foundation becomes zero since there exist no transmitted excitation forces to the foundation from the AB-rotor unit due to the balancing capability of AB for an imbalance rotor. Therefore, from the result of Figure 3, it is seen that the balancing performance of AB can be successfully achieved for such coupled system.


### 6.2. Validation of Analytical Limit Cycle Solutions

In Figure4, the comparisons between the steady-state analytical limit cycle solutions(LC) and the numerical simulations(Num) are considered. For simplicity, the short notationsLC andNum are utilized throughout this paper. The notationLC represents the simulation outcomes for analytical solution (19a), (19b), and (19c). On the other hands, the notationNum indicates the results obtained through the full nonlinear simulation of (10). Based upon [13], for the simulation ofNum, the initial positions of two balls are assumed to be merged together such as ϕ1(0)=ϕ2(0)=0∘, which is more prominent initial condition to generate the limit cycle response. The results in (a-1) through (a-4) of Figure 4 are obtained at Ω-=1.27. And the outcomes in (b-1) through (b-4) are simulated at Ω-=2.93 while the findings in (c-1) through (c-4) are produced at Ω-=3.79.Figure 4
Comparison between analytical limit cycle solution and numerical simulation on time domain ((a-1) through (a-4) atΩ-=1.27, (b-1) through (b-4) at Ω-=2.93, and (c-1) through (c-4) at Ω-=3.79).Under the limit cycle condition, the outcomes in (a-1), (b-1), and (c-1) exhibit the relative whirl speeds of merged balls circulating around the rotating rotor (i.e., nonsynchronous manner). While (a-2), (a-3), (b-2), (b-3), (c-2), and (c-3) represent the steady-state transversal deflections of rotor, (a-4), (b-4), and (c-4) indicate the corresponding vertical deflections of foundation. It is found from the results of Figure4 that the proposed solution in (19a), (19b), and (19c) is perfectly synchronized with the responses ofNum, and the whirl speeds ω ofLC capture the average value of whirl speeds inNum. Since ω ofLC is assumed to be a constant, it is obvious that the analytical solution ω is not able to characterize the insignificantly small oscillation ofNum. Furthermore, (a) through (c) of Figure 5 closely investigate the coupled oscillating behavior (i.e., y(t)  and  yf(t)) of two primary components (i.e., AB/rotor unit and foundation). And the whirl orbits (i.e., x(t) versus y(t)) of rotor have been depicted in (d) of Figure 5.Figure 5
Coupled oscillating behavior of rotor and foundation and rotor whirl orbits under limit cycle condition ((a) atΩ-=1.27, (b) at Ω-=2.93, and (c) at Ω-=3.79).
(a) (b) (c) (d)As shown in (a), (b), and (c) of Figure5, in direction-wise oscillating of rotor and foundation, the vertical deflection of rotor is synchronized with foundation’s at Ω-=1.27, while the foundation relatively moves in the opposite direction of rotor at higher speeds Ω-=2.93 and Ω-=3.79. This behavior is, in some degree, similar to the well-known first and second normal modes of the two-mass vertically coupled system via two springs.Also, compared to the responses of single AB/rotor unit with a symmetric support, the outcome in (d) of Figure5 shows us that, due to vertically coupled configuration, the shape of whirl orbits is elliptical and several different whirling behavior patterns are also observed according to different operating speeds. Specifically, the rotor is majorly horizontally oscillated at Ω-=1.27 and primarily vertically shook at Ω-=2.93. On the other hand, the rotor is subject to the tilted whirl orbit at Ω-=3.79.Apparently, the findings in Figure5 provide the new insight about the responses of AB/rotor/bearing system coupled with another vibrating subsystem such as flexible foundation.Furthermore, Figure6 addresses the extensive comparisons of steady-state responses between analytical solutions(LC) and numerical simulation(Num) at every operating speed in Ω-∈07.Figure 6
Extensive comparison between analytical solution(LC) and numerical simulation(Num) with the inherent nature of whirling speed of AB under limit cycles.
(a) (b) (c) (d) (e)To see how to obtain the results ofNum in Figure 6, the computation method ofNum should be explained before proceeding. The subscript N has been added to each variable ofNum to be distinguished from the notation of analytical solutions(LC). The average whirl speed ofNum presented in (a) of Figure 6 is computed by(25)ωave.N=0.5maxT-ta<t<Tϕ˙Nt+minT-ta<t<Tϕ˙Nt.At each operating speed, the maximum whirl amplitudes of both rotor and foundation depicted in (b) through (d) of Figure 6 are, respectively, calculated by maxT-ta<t<T⁡xN(t), maxT-ta<t<T⁡yN(t), and maxT-ta<t<T⁡yfN(t), where T and ta indicate the end of simulation time and the interval of sampling time, respectively.Now, let us consider the outcomes in (a) of Figure6 showing that the whirl speed ofLC is well synchronized with the average whirl speed ofNum (i.e., ωave.N). Along the rotor speeds, the three separate branches (denoted asBr.1,Br.2, andBr.3) of limit cycle are exhibited, which are bifurcated at near critical speeds Ω-cr.1, Ω-cr.2, and Ω-cr.3. Specifically, it is observed that the nonsynchronous limit cycle conditions satisfying ω≠0 are initiated at near each critical speed, prolonged on corresponding supercritical speeds or up to the next ones, and finally terminated at a certain point. Before proceeding, let us revisit to the results in Figure 5. Here, it should be mentioned that each response separately obtained at Ω-=1.27, Ω-=2.93, and Ω-=3.79 in Figure 5 corresponds to one of the solutions indicated by the dotted rectangular boxes in (a) of Figure 6 and belongs to each branch of limit cycle. Therefore, recalling from the outcomes of Figure 5 that the moving direction of rotor relative to foundation is altered and the shape of rotor whirl orbit is transformed as the system experiences each critical speed along the rotor speeds, we can see that the limit cycle behavior of each branch is characterized by the inherent nature of each critical speed (i.e., each mode of system). In (b) and (c) of Figure 6, the maximum values of rotor displacements along the speeds are, respectively, shown and it is found that the maximum values, A12+A22 and B12+B22, of LC well agree with the maximum amplitudes of Num, maxT-ta<t<T⁡xN(t) and maxT-ta<t<T⁡yN(t), in every operating speed except near critical speeds. Specifically, some of Num values near Ω-cr.1, Ω-cr.2, and Ω-cr.3 are slightly fallen apart from the limit cycle solutions (LC).In (d) of Figure6, the maximum vertical amplitudes of foundation are depicted. Again, the amplitudes of LC are obtained via C12+C22 and are precisely able to capture the maximum values of Num (i.e., maxT-ta<t<T⁡yfN(t)) except for near critical speeds.Therefore, it is observed from Figure6 that the proposed analytical steady-state solutions in (19a), (19b), and (19c) are accurate enough to capture the nonsynchronous limit cycle responses of AB/rotor/foundation system over the entire of operating speeds except for some cases near critical speeds. We will shortly discuss some of these interesting behavior patterns near resonances in Figure 7.Figure 7
System behavior near resonance (b) atΩ-=2.14, (c) Ω-=2.26, and (d) Ω-=2.42.
(a) (b) (c) (d)Additionally, in (e) of Figure6, the whirl speeds in (a) of Figure 6 are expressed with reference to {n1,n2,n3} and it shows that they are almost synchronized with the critical speeds Ω-cr.1, Ω-cr.2, and Ω-cr.3. This implies that the excitation frequency of force term Fball in (10) or (13) is matched with the critical speeds (natural frequencies of nominal system without AB) inducing a large whirling amplitude of system, which represents the essence of undesirable limit cycle condition.Now, let us explore the eccentricity between Num and LC displayed near critical speeds as shown in (b), (c), and (d) of Figure6. Thus, Figure 7 revisits the results in (b) of Figure 6 and closely investigates the behavior of system near Ω-cr.2. It is found that the discrepancy between Num (i.e., maxT-ta<t<T⁡xN(t)) and A12+A22 is exhibited in the vicinity of Ω-cr.2 and disappeared as Ω- increases. This is caused by the effect of synchronous portion A32+A42  which is not negligible near Ω-cr.2. However, as shown in (a) of Figure 7, the nonsynchronous amplitude, A12+A22, becomes dominant over the synchronous amplitude A32+A42 along 2.26<Ω-<3.8 except near Ω-cr.2. Since the limit cycle solutions in (19a), (19b), and (19c) do not consider the synchronous portion, they are not able to capture the response of system near resonances. However, the effect of synchronous responses becomes insignificant along the corresponding supercritical speeds right above resonances. This point of view can be more clearly seen from the time domain simulations shown in (b), (c), and (d) of Figure 7. Here, (b) of Figure 7 indicates the response of system at resonance (Ω-=Ω-cr.2=2.14) while the other two results (i.e., (c) and (d) of Figure 7) do the behavior at speeds, Ω-=2.26 and Ω-=2.42, slightly higher than resonance. The notations xsyn.ss(t) and xL.ss(t) used in Figure 7, respectively, represent the synchronous part (i.e., xsyn.ss(t)=A3cos(Ωt)+A4sin(Ωt)) due to imbalance and the nonsynchronous parts in (19a) via AB. x(t) is obtained by xsyn.ss(t)+xL.ss(t). As seen in (b), (c), and (d) of Figure 7, it is verified that the effect of the synchronous part, xsyn.ss(t), is considerable at the resonance but becomes quite negligible above the resonance. Therefore, it is found from (c) and (d) of Figure 7 that x(t)≈xL.ss(t) along the supercritical speeds above Ω-cr.2. Also, it is observed from (b) of Figure 7 that the system behavior at the resonance is understood by the combination of xsyn.ss(t) and xL.ss(t).


### 6.3. Balancing and Limit Cycle Stability Results

If the limit cycle of AB can be destabilized by any means, the stable balancing performance at the supercritical speeds can be completely guaranteed. One of the primary parameters to do this is the system damping; thus it is considered as a primary design parameter in this paper and is explored here.Since the effect ofCr directly providing the damping force to the AB/rotor is well studied for limit cycles in several literatures, it is very important for us to investigate the coupled damping effect Cf for the limit cycle response of system proposed here.In Figure8, the limit cycle behavior has been explored over the speed ranges Ω-∈06.5 for given three different damping values Cf/Cr=0.7, Cf/Cr=1.05, and Cf/Cr=2.1 with Cr=0.001Kr.Figure 8
Effect of damping ratioCf/Cr and rotor speed ratio Ω- for limit cycle behavior.
(a) (b) (c) (d) (e) (f)In (a) of Figure8, the straight lines with stars represent the critical speeds and S1, S2, and S3, respectively, indicate the ranges of 1st, 2nd, and 3rd supercritical speeds. Before studying the effect of Cf, let us take a look at (e) and (f) of Figure 8 which zoom in the case with Cf/Cr=2.1 from (a) of Figure 8. According to Floquet stability criterion in Section 5, it is determined that the responses depicted as straight lines in (e) and (f) of Figure 8 represent physically feasible limit cycles while the dotted lines are not. This implies that solutions obtained by Section 4 are not all stable. It is found that the solutions corresponding to the portions from each subcritical speed (i.e., the speeds below each critical speed) to each turning point indicated by a red circle are stable. Furthermore, these stable solutions can be classified into two types of response via the criterion whether the relative whirling speed is zero ω=0 or not ω≠0. Specifically, the condition with ω=0 appears in each subcritical speed and is considered as a synchronous merged equilibrium where two balls are merged together and synchronized with rotor rotating speed Ω, but the two merged balls tend to stay at near angular phase of imbalance ϕim; thus it increases the imbalances of system [1–10], which is also unfavorable action of AB emerging in subcritical speeds. On the other hand, needless to say, the portions with nonsynchronous condition ω≠0 indicate the unwanted limit cycle condition coexisting with stable balanced condition at supercritical speeds.Now returning to the investigation of the damping effect, we can see that the increase inCf leads to a less dominance of limit cycles persistence in both the 1st and 3rd supercritical speeds. Also, considering the tendency of vertical amplitudes,y(t) (i.e., B12+B22) and yf(t) (i.e., C12+C22), of rotor and foundation shown in (c) and (d) of Figure 8, it clearly supports the possibility that the damping effect of flexible foundation Cf/Cr can be one of the primary design parameters that influence the whirling magnitude and persistence. However, it is unfortunately found that the effect of Cf has no influence on the limit cycle bifurcated near the 2nd critical speed, which represents the horizontal amplitude of rotor x(t) (i.e., A12+A22) not directly interacting with Cf. Additionally, it is shown from (a) of Figure 8 that the multiple limit cycles coexist in the regime of 3rd supercritical speed, where the limit cycle bifurcated near the 2nd critical speed is persistent along the 2nd and 3rd supercritical speeds and coexists with the limit cycle originated near the 3rd critical speed. This indicates the undesirable coexistence of two limit cycle responses with a balanced one in the ranges of 3rd supercritical speed.Now, to perform the extensive investigation for the effect of dampingCr, the stable regions about the balanced, merged, and limit cycle responses as a function of Cr at two different values of mb along the speeds have been presented in Figure 9. Before proceeding further, for simplicity, the short notations utilized in Figure 8 need to be explained below and will be used throughout the paper.(1)
L1, L2, L3: nonsynchronous limit cycle regions about (19a), (19b), and (19c), respectively, bifurcated at the 1st, 2nd, and 3rd critical speeds and satisfying ω≠0.(2)
B1, B2, B3: stable balanced regions about (11a), (11b), and (11c), respectively, emerged at the 1st, 2nd, and 3rd supercritical speeds.(3)
M1, M2, M3: stable regions about the merged response, respectively, appeared at the 1st, 2nd, and 3rd subcritical speeds and satisfied ω=0, which have been discussed from the results in both (e) and (f) of Figure 8.Figure 9
Effect of damping ratioCf/Cr and rotor speed ratio Ω- on stable balanced, merged, and limit cycle regions for (a) mb=0.002kg and (b) mb=0.003 kg.
(a) (b)From Figure9, the undesired coexistence of limit cycle(s) and balanced condition is observed in each supercritical speed such as B1+L1, B2+L2, B3+L2+L3, B3+L3 and B3+L2, where the system can be attracted into either undesirable condition(s) or a wanted one. This coexistence behavior is determined based on the initial condition of two balls, rotor speed, and bearing support [12, 13, 20–23].By exploring the results shown in Figure9, it is clearly seen that the coexistence regimes become less prominent as Cf/Cr is increased and mb becomes smaller. However, as shown in Figure 9, the variation of Cf/Cr hardly affects the region L2, which has been also observed from the outcomes in (b) of Figure 8.In Figure9, the effect of Cr/Cf has been investigated along the rotor speeds with two different mb.Compared to the effect ofCf/Cr shown in Figure 9, it is found from Figure 10 that the effect of Cr/Cf is able to deliver the impact to all limit cycle regions L1, L2, and L3, which is well understood in AB researches and matched with the outcomes [23] that all the limit cycles over operating speeds are affected by bearing damping directly supporting the rotor.Figure 10
Effect of damping ratioCr/Cf and rotor speed ratio Ω- on stable balanced, merged, and limit cycle regions for (a) mb=0.002kg and (b) mb=0.003kg.
(a) (b)By seeing the different aspect about L2 from both Figures9 and 10 for damping effect, in order to suppress all possible limit cycles, it is apparent that tuning damping Cr/Cf directly interacting with the source of limit cycle (i.e., AB/rotor) is the primary choice. However, if the Cr/Cf cannot be tunable for a given circumstance, the appropriate selection of Cf/Cr is the alternative mean to mitigate the limit cycle in some degrees.Therefore, due to coupled oscillating feature introduced in this paper, it is newly found that the damping associated component not directly attached to the AB/rotor unit but coupled through another vibrating subsystem can also affect the response of limit cycle. This result can be a valuable asset widely employed into the autobalancer application of the rotor system dynamically interacting with another oscillating mechanical subsystem via a spring-damper based flexible support.Additionally, Figure11 investigates the stable regions of all responses for the effect of stiffness Kf with Kr=1000 N/m. As shown in (a) of Figure 11, as the stiffness, Kf, of foundation is varied, the critical speeds of system are accordingly shifted; thus the origins of limit cycles L1 and L3 directly influenced by Kf are transferred along the speeds. Also, the augmentation of Kf leads the impact of L1 to become dominated, but the region of L3 has been reduced. Similarly, as seen from the effect of Cf in Figure 9, the effect of L2 not directly related to Kf is not affected via the variation of Kf. Due to the effect of Kf, as shown in (b) of Figure 11, it is found that other coexistence such as M3+L1+L2+L3, B3+L1+L2+L3, M2+L1+L2, M3+L2+L3, and B2+L1+L2 not accounted from Figures 9 and 10 emerged. Here, there is even coexistence of four different responses, B3+L1+L2+L3, which is described by three limit cycles and a stable balanced condition along the 3rd supercritical speeds. From the outcomes of Figure 11, it clearly shows that the indirectly associated stiffness is also an important parameter to govern the undesirable behavior of given AB/rotor/foundation system.Figure 11
Effect of stiffnessKf and rotor speed Ω on stable balanced, merged, and limit cycle regions.
(a) (b)Therefore, as the results in Figures9, 10, and 11 are discussed so far, due to such special configuration described here, the multiple limit cycles are encountered in the range of supercritical speeds and more complicated coexistence has been attracted into the system. And, it is found that the operating speed and appropriate system parameters should be cautiously selected to destabilize the undesirable behavior of such coupled system and guarantee the stable balanced response of AB, compared to the well-known responses of single unit AB/rotor system.


## 6.1. Balanced Response

With the default valueKf=700 N/m, Figure 3 shows the balanced responses on the time domain obtained via (10) based upon the initial conditions of two balls ϕ1(0)=0∘ and ϕ2(0)=180∘ at the three different speeds Ω-=1.25, Ω-=2.58, and Ω-=4.12. Specifically, the results in (a-1) through (a-3) of Figure 3 are obtained at the operating speed Ω-=1.25. And the outcomes in (b-1) through (b-3) are simulated at Ω-=2.58 while the findings in (c-1) through (c-3) are captured at Ω-=4.12. Beside balanced responses, the responses of nominal system (without AB) are presented in Figure 3 and compared with the balancing performances.Figure 3
Balancing responses of system on time domain ((a-1) through (a-3) atΩ-=1.25, (b-1) through (b-3) at Ω-=2.58, and (c-1) through (c-3) at Ω-=4.12).Based onKf=700 N/m and other parameter values in Table 1, we can find that the nondimensional critical speeds are Ω-cr.1=1, Ω-cr.2=2.17, and Ω-cr.3=2.75. Therefore, it is found that (a-1) through (a-3), (b-1) through (b-3), and (c-1) through (c-3), respectively, indicate the balancing responses emerging on the 1st supercritical speeds (i.e., Ω-cr.1=1<Ω-<Ω-cr.2=2.17), the 2nd supercritical speeds (i.e., Ω-cr.2=2.17<Ω-<Ω-cr.3=2.75), and the 3rd supercritical speeds (i.e., Ω->Ω-cr.3=2.75), which are well matched with the outcomes in [16] that the balancing responses are found in each supercritical speed.Moreover, it is observed that the final angular phases of two balls shown in (a-3), (b-3), and (c-3) are identical to the steady-state angular positions of balls, analytically computed based on (11b). The AB parameters used here are mb=0.002 kg and rb=0.3. And the imbalance parameters are mim=0.002 kg and rim=0.3 with ϕim=45∘. Consequently, for certain operating conditions, we can see that the action of AB achieves the perfect balancing of eccentric rotor resulting in zero vertical deflection of foundation as well. Here, it is so obvious that the vertical deflection of foundation becomes zero since there exist no transmitted excitation forces to the foundation from the AB-rotor unit due to the balancing capability of AB for an imbalance rotor. Therefore, from the result of Figure 3, it is seen that the balancing performance of AB can be successfully achieved for such coupled system.


## 6.2. Validation of Analytical Limit Cycle Solutions

In Figure4, the comparisons between the steady-state analytical limit cycle solutions(LC) and the numerical simulations(Num) are considered. For simplicity, the short notationsLC andNum are utilized throughout this paper. The notationLC represents the simulation outcomes for analytical solution (19a), (19b), and (19c). On the other hands, the notationNum indicates the results obtained through the full nonlinear simulation of (10). Based upon [13], for the simulation ofNum, the initial positions of two balls are assumed to be merged together such as ϕ1(0)=ϕ2(0)=0∘, which is more prominent initial condition to generate the limit cycle response. The results in (a-1) through (a-4) of Figure 4 are obtained at Ω-=1.27. And the outcomes in (b-1) through (b-4) are simulated at Ω-=2.93 while the findings in (c-1) through (c-4) are produced at Ω-=3.79.Figure 4
Comparison between analytical limit cycle solution and numerical simulation on time domain ((a-1) through (a-4) atΩ-=1.27, (b-1) through (b-4) at Ω-=2.93, and (c-1) through (c-4) at Ω-=3.79).Under the limit cycle condition, the outcomes in (a-1), (b-1), and (c-1) exhibit the relative whirl speeds of merged balls circulating around the rotating rotor (i.e., nonsynchronous manner). While (a-2), (a-3), (b-2), (b-3), (c-2), and (c-3) represent the steady-state transversal deflections of rotor, (a-4), (b-4), and (c-4) indicate the corresponding vertical deflections of foundation. It is found from the results of Figure4 that the proposed solution in (19a), (19b), and (19c) is perfectly synchronized with the responses ofNum, and the whirl speeds ω ofLC capture the average value of whirl speeds inNum. Since ω ofLC is assumed to be a constant, it is obvious that the analytical solution ω is not able to characterize the insignificantly small oscillation ofNum. Furthermore, (a) through (c) of Figure 5 closely investigate the coupled oscillating behavior (i.e., y(t)  and  yf(t)) of two primary components (i.e., AB/rotor unit and foundation). And the whirl orbits (i.e., x(t) versus y(t)) of rotor have been depicted in (d) of Figure 5.Figure 5
Coupled oscillating behavior of rotor and foundation and rotor whirl orbits under limit cycle condition ((a) atΩ-=1.27, (b) at Ω-=2.93, and (c) at Ω-=3.79).
(a) (b) (c) (d)As shown in (a), (b), and (c) of Figure5, in direction-wise oscillating of rotor and foundation, the vertical deflection of rotor is synchronized with foundation’s at Ω-=1.27, while the foundation relatively moves in the opposite direction of rotor at higher speeds Ω-=2.93 and Ω-=3.79. This behavior is, in some degree, similar to the well-known first and second normal modes of the two-mass vertically coupled system via two springs.Also, compared to the responses of single AB/rotor unit with a symmetric support, the outcome in (d) of Figure5 shows us that, due to vertically coupled configuration, the shape of whirl orbits is elliptical and several different whirling behavior patterns are also observed according to different operating speeds. Specifically, the rotor is majorly horizontally oscillated at Ω-=1.27 and primarily vertically shook at Ω-=2.93. On the other hand, the rotor is subject to the tilted whirl orbit at Ω-=3.79.Apparently, the findings in Figure5 provide the new insight about the responses of AB/rotor/bearing system coupled with another vibrating subsystem such as flexible foundation.Furthermore, Figure6 addresses the extensive comparisons of steady-state responses between analytical solutions(LC) and numerical simulation(Num) at every operating speed in Ω-∈07.Figure 6
Extensive comparison between analytical solution(LC) and numerical simulation(Num) with the inherent nature of whirling speed of AB under limit cycles.
(a) (b) (c) (d) (e)To see how to obtain the results ofNum in Figure 6, the computation method ofNum should be explained before proceeding. The subscript N has been added to each variable ofNum to be distinguished from the notation of analytical solutions(LC). The average whirl speed ofNum presented in (a) of Figure 6 is computed by(25)ωave.N=0.5maxT-ta<t<Tϕ˙Nt+minT-ta<t<Tϕ˙Nt.At each operating speed, the maximum whirl amplitudes of both rotor and foundation depicted in (b) through (d) of Figure 6 are, respectively, calculated by maxT-ta<t<T⁡xN(t), maxT-ta<t<T⁡yN(t), and maxT-ta<t<T⁡yfN(t), where T and ta indicate the end of simulation time and the interval of sampling time, respectively.Now, let us consider the outcomes in (a) of Figure6 showing that the whirl speed ofLC is well synchronized with the average whirl speed ofNum (i.e., ωave.N). Along the rotor speeds, the three separate branches (denoted asBr.1,Br.2, andBr.3) of limit cycle are exhibited, which are bifurcated at near critical speeds Ω-cr.1, Ω-cr.2, and Ω-cr.3. Specifically, it is observed that the nonsynchronous limit cycle conditions satisfying ω≠0 are initiated at near each critical speed, prolonged on corresponding supercritical speeds or up to the next ones, and finally terminated at a certain point. Before proceeding, let us revisit to the results in Figure 5. Here, it should be mentioned that each response separately obtained at Ω-=1.27, Ω-=2.93, and Ω-=3.79 in Figure 5 corresponds to one of the solutions indicated by the dotted rectangular boxes in (a) of Figure 6 and belongs to each branch of limit cycle. Therefore, recalling from the outcomes of Figure 5 that the moving direction of rotor relative to foundation is altered and the shape of rotor whirl orbit is transformed as the system experiences each critical speed along the rotor speeds, we can see that the limit cycle behavior of each branch is characterized by the inherent nature of each critical speed (i.e., each mode of system). In (b) and (c) of Figure 6, the maximum values of rotor displacements along the speeds are, respectively, shown and it is found that the maximum values, A12+A22 and B12+B22, of LC well agree with the maximum amplitudes of Num, maxT-ta<t<T⁡xN(t) and maxT-ta<t<T⁡yN(t), in every operating speed except near critical speeds. Specifically, some of Num values near Ω-cr.1, Ω-cr.2, and Ω-cr.3 are slightly fallen apart from the limit cycle solutions (LC).In (d) of Figure6, the maximum vertical amplitudes of foundation are depicted. Again, the amplitudes of LC are obtained via C12+C22 and are precisely able to capture the maximum values of Num (i.e., maxT-ta<t<T⁡yfN(t)) except for near critical speeds.Therefore, it is observed from Figure6 that the proposed analytical steady-state solutions in (19a), (19b), and (19c) are accurate enough to capture the nonsynchronous limit cycle responses of AB/rotor/foundation system over the entire of operating speeds except for some cases near critical speeds. We will shortly discuss some of these interesting behavior patterns near resonances in Figure 7.Figure 7
System behavior near resonance (b) atΩ-=2.14, (c) Ω-=2.26, and (d) Ω-=2.42.
(a) (b) (c) (d)Additionally, in (e) of Figure6, the whirl speeds in (a) of Figure 6 are expressed with reference to {n1,n2,n3} and it shows that they are almost synchronized with the critical speeds Ω-cr.1, Ω-cr.2, and Ω-cr.3. This implies that the excitation frequency of force term Fball in (10) or (13) is matched with the critical speeds (natural frequencies of nominal system without AB) inducing a large whirling amplitude of system, which represents the essence of undesirable limit cycle condition.Now, let us explore the eccentricity between Num and LC displayed near critical speeds as shown in (b), (c), and (d) of Figure6. Thus, Figure 7 revisits the results in (b) of Figure 6 and closely investigates the behavior of system near Ω-cr.2. It is found that the discrepancy between Num (i.e., maxT-ta<t<T⁡xN(t)) and A12+A22 is exhibited in the vicinity of Ω-cr.2 and disappeared as Ω- increases. This is caused by the effect of synchronous portion A32+A42  which is not negligible near Ω-cr.2. However, as shown in (a) of Figure 7, the nonsynchronous amplitude, A12+A22, becomes dominant over the synchronous amplitude A32+A42 along 2.26<Ω-<3.8 except near Ω-cr.2. Since the limit cycle solutions in (19a), (19b), and (19c) do not consider the synchronous portion, they are not able to capture the response of system near resonances. However, the effect of synchronous responses becomes insignificant along the corresponding supercritical speeds right above resonances. This point of view can be more clearly seen from the time domain simulations shown in (b), (c), and (d) of Figure 7. Here, (b) of Figure 7 indicates the response of system at resonance (Ω-=Ω-cr.2=2.14) while the other two results (i.e., (c) and (d) of Figure 7) do the behavior at speeds, Ω-=2.26 and Ω-=2.42, slightly higher than resonance. The notations xsyn.ss(t) and xL.ss(t) used in Figure 7, respectively, represent the synchronous part (i.e., xsyn.ss(t)=A3cos(Ωt)+A4sin(Ωt)) due to imbalance and the nonsynchronous parts in (19a) via AB. x(t) is obtained by xsyn.ss(t)+xL.ss(t). As seen in (b), (c), and (d) of Figure 7, it is verified that the effect of the synchronous part, xsyn.ss(t), is considerable at the resonance but becomes quite negligible above the resonance. Therefore, it is found from (c) and (d) of Figure 7 that x(t)≈xL.ss(t) along the supercritical speeds above Ω-cr.2. Also, it is observed from (b) of Figure 7 that the system behavior at the resonance is understood by the combination of xsyn.ss(t) and xL.ss(t).


## 6.3. Balancing and Limit Cycle Stability Results

If the limit cycle of AB can be destabilized by any means, the stable balancing performance at the supercritical speeds can be completely guaranteed. One of the primary parameters to do this is the system damping; thus it is considered as a primary design parameter in this paper and is explored here.Since the effect ofCr directly providing the damping force to the AB/rotor is well studied for limit cycles in several literatures, it is very important for us to investigate the coupled damping effect Cf for the limit cycle response of system proposed here.In Figure8, the limit cycle behavior has been explored over the speed ranges Ω-∈06.5 for given three different damping values Cf/Cr=0.7, Cf/Cr=1.05, and Cf/Cr=2.1 with Cr=0.001Kr.Figure 8
Effect of damping ratioCf/Cr and rotor speed ratio Ω- for limit cycle behavior.
(a) (b) (c) (d) (e) (f)In (a) of Figure8, the straight lines with stars represent the critical speeds and S1, S2, and S3, respectively, indicate the ranges of 1st, 2nd, and 3rd supercritical speeds. Before studying the effect of Cf, let us take a look at (e) and (f) of Figure 8 which zoom in the case with Cf/Cr=2.1 from (a) of Figure 8. According to Floquet stability criterion in Section 5, it is determined that the responses depicted as straight lines in (e) and (f) of Figure 8 represent physically feasible limit cycles while the dotted lines are not. This implies that solutions obtained by Section 4 are not all stable. It is found that the solutions corresponding to the portions from each subcritical speed (i.e., the speeds below each critical speed) to each turning point indicated by a red circle are stable. Furthermore, these stable solutions can be classified into two types of response via the criterion whether the relative whirling speed is zero ω=0 or not ω≠0. Specifically, the condition with ω=0 appears in each subcritical speed and is considered as a synchronous merged equilibrium where two balls are merged together and synchronized with rotor rotating speed Ω, but the two merged balls tend to stay at near angular phase of imbalance ϕim; thus it increases the imbalances of system [1–10], which is also unfavorable action of AB emerging in subcritical speeds. On the other hand, needless to say, the portions with nonsynchronous condition ω≠0 indicate the unwanted limit cycle condition coexisting with stable balanced condition at supercritical speeds.Now returning to the investigation of the damping effect, we can see that the increase inCf leads to a less dominance of limit cycles persistence in both the 1st and 3rd supercritical speeds. Also, considering the tendency of vertical amplitudes,y(t) (i.e., B12+B22) and yf(t) (i.e., C12+C22), of rotor and foundation shown in (c) and (d) of Figure 8, it clearly supports the possibility that the damping effect of flexible foundation Cf/Cr can be one of the primary design parameters that influence the whirling magnitude and persistence. However, it is unfortunately found that the effect of Cf has no influence on the limit cycle bifurcated near the 2nd critical speed, which represents the horizontal amplitude of rotor x(t) (i.e., A12+A22) not directly interacting with Cf. Additionally, it is shown from (a) of Figure 8 that the multiple limit cycles coexist in the regime of 3rd supercritical speed, where the limit cycle bifurcated near the 2nd critical speed is persistent along the 2nd and 3rd supercritical speeds and coexists with the limit cycle originated near the 3rd critical speed. This indicates the undesirable coexistence of two limit cycle responses with a balanced one in the ranges of 3rd supercritical speed.Now, to perform the extensive investigation for the effect of dampingCr, the stable regions about the balanced, merged, and limit cycle responses as a function of Cr at two different values of mb along the speeds have been presented in Figure 9. Before proceeding further, for simplicity, the short notations utilized in Figure 8 need to be explained below and will be used throughout the paper.(1)
L1, L2, L3: nonsynchronous limit cycle regions about (19a), (19b), and (19c), respectively, bifurcated at the 1st, 2nd, and 3rd critical speeds and satisfying ω≠0.(2)
B1, B2, B3: stable balanced regions about (11a), (11b), and (11c), respectively, emerged at the 1st, 2nd, and 3rd supercritical speeds.(3)
M1, M2, M3: stable regions about the merged response, respectively, appeared at the 1st, 2nd, and 3rd subcritical speeds and satisfied ω=0, which have been discussed from the results in both (e) and (f) of Figure 8.Figure 9
Effect of damping ratioCf/Cr and rotor speed ratio Ω- on stable balanced, merged, and limit cycle regions for (a) mb=0.002kg and (b) mb=0.003 kg.
(a) (b)From Figure9, the undesired coexistence of limit cycle(s) and balanced condition is observed in each supercritical speed such as B1+L1, B2+L2, B3+L2+L3, B3+L3 and B3+L2, where the system can be attracted into either undesirable condition(s) or a wanted one. This coexistence behavior is determined based on the initial condition of two balls, rotor speed, and bearing support [12, 13, 20–23].By exploring the results shown in Figure9, it is clearly seen that the coexistence regimes become less prominent as Cf/Cr is increased and mb becomes smaller. However, as shown in Figure 9, the variation of Cf/Cr hardly affects the region L2, which has been also observed from the outcomes in (b) of Figure 8.In Figure9, the effect of Cr/Cf has been investigated along the rotor speeds with two different mb.Compared to the effect ofCf/Cr shown in Figure 9, it is found from Figure 10 that the effect of Cr/Cf is able to deliver the impact to all limit cycle regions L1, L2, and L3, which is well understood in AB researches and matched with the outcomes [23] that all the limit cycles over operating speeds are affected by bearing damping directly supporting the rotor.Figure 10
Effect of damping ratioCr/Cf and rotor speed ratio Ω- on stable balanced, merged, and limit cycle regions for (a) mb=0.002kg and (b) mb=0.003kg.
(a) (b)By seeing the different aspect about L2 from both Figures9 and 10 for damping effect, in order to suppress all possible limit cycles, it is apparent that tuning damping Cr/Cf directly interacting with the source of limit cycle (i.e., AB/rotor) is the primary choice. However, if the Cr/Cf cannot be tunable for a given circumstance, the appropriate selection of Cf/Cr is the alternative mean to mitigate the limit cycle in some degrees.Therefore, due to coupled oscillating feature introduced in this paper, it is newly found that the damping associated component not directly attached to the AB/rotor unit but coupled through another vibrating subsystem can also affect the response of limit cycle. This result can be a valuable asset widely employed into the autobalancer application of the rotor system dynamically interacting with another oscillating mechanical subsystem via a spring-damper based flexible support.Additionally, Figure11 investigates the stable regions of all responses for the effect of stiffness Kf with Kr=1000 N/m. As shown in (a) of Figure 11, as the stiffness, Kf, of foundation is varied, the critical speeds of system are accordingly shifted; thus the origins of limit cycles L1 and L3 directly influenced by Kf are transferred along the speeds. Also, the augmentation of Kf leads the impact of L1 to become dominated, but the region of L3 has been reduced. Similarly, as seen from the effect of Cf in Figure 9, the effect of L2 not directly related to Kf is not affected via the variation of Kf. Due to the effect of Kf, as shown in (b) of Figure 11, it is found that other coexistence such as M3+L1+L2+L3, B3+L1+L2+L3, M2+L1+L2, M3+L2+L3, and B2+L1+L2 not accounted from Figures 9 and 10 emerged. Here, there is even coexistence of four different responses, B3+L1+L2+L3, which is described by three limit cycles and a stable balanced condition along the 3rd supercritical speeds. From the outcomes of Figure 11, it clearly shows that the indirectly associated stiffness is also an important parameter to govern the undesirable behavior of given AB/rotor/foundation system.Figure 11
Effect of stiffnessKf and rotor speed Ω on stable balanced, merged, and limit cycle regions.
(a) (b)Therefore, as the results in Figures9, 10, and 11 are discussed so far, due to such special configuration described here, the multiple limit cycles are encountered in the range of supercritical speeds and more complicated coexistence has been attracted into the system. And, it is found that the operating speed and appropriate system parameters should be cautiously selected to destabilize the undesirable behavior of such coupled system and guarantee the stable balanced response of AB, compared to the well-known responses of single unit AB/rotor system.


## 7. Concluding Remarks

This research primarily focuses on the nonlinear and automatic balancing behavior of autobalancer (AB) system for a planar eccentric rotor coupled with the foundation structure vertically supported by damping-spring type isolator. This kind of setting can be possibly applicable to the vibration isolation problem for many mechanical systems containing the inherent imbalance natured rotor such as a washing machine or a compressor with a flexible foundation. To explore the behavior of this special arrangement never been previously considered in AB researches, solutions for the stable synchronous balanced response and the nonsynchronous limit cycle of given system have been, respectively, obtained via a fixed equilibrium and a harmonic balancing like approach. Furthermore, the stability of each response is assessed via a perturbation and Floquet analysis, and the undesirable coexistence of the stable balanced synchronous and undesirable nonsynchronous limit cycle as a function of the system parameters and operating speeds has been thoroughly investigated. Based on investigation performed in this paper, we found the following:(1)
The system presented here is characterized by three critical speeds; thus the stable synchronous balanced behavior (i.e., B1, B2, B3) of system separately appears in each supercritical speed in between each critical speed (i.e.,Ω-cr.1, Ω-cr.2, and Ω-cr.3) and three separate limit cycle branches (i.e., L1, L2, L3) bifurcated at each critical speed are also exhibited.(2)
According to given operating speeds, it is observed that, under the limit cycle condition, the relative moving direction of coupled objects (i.e., rotor + foundation) and the shape of whirl orbit for rotor are varied. Specifically, according to givenΩ-, both objects move in identical direction or opposite direction and the shape of whirl orbit appears as several different elliptical trajectories. This behavior is, in some degree, similar to the well-known first and second normal modes of the two-mass vertically coupled system via two springs.(3)
Due to coupled configuration described here, the multiple nonsynchronous limit cycles are encountered in the range of supercritical speeds and more complicated coexistence is invited into the system such as B1+L1, B2+L2, B3+L2+L3, B3+L3, B2+L1+L2, and B3+L2 where the balanced behavior and one or multiple limit cycle conditions coexist together.(4)
Such undesirable coexistence is heavily affected by the damping ratiosCf/Cr and Cr/Cf. It is needless to say that, in order to suppress all limit cycles appearing along the rotor speed, selecting the appropriate damping Cr directly connected with the source of limit cycle (i.e., AB + rotor) is the primary design choice. This is well revealed from existing AB researches. However, if Cr cannot be tunable for a given circumstance, the appropriate selection of Cf would be the alternative mean to mitigate the limit cycle in some degrees. It is observed that the effect of stiffness Kf also influences the limit cycles. Interestingly, as Kf is increased, the origin of limit cycles L1 and L3 directly influenced by Kf is transferred to higher speeds and the impact of L1 becomes dominated, but the region of L3 has been attenuated. Therefore, it is found that, due to coupled feature of system, the damping and stiffness components not directly associated with the AB/rotor unit but coupled through another vibrating subsystem can also affect the response of limit cycle.Considering the dynamically coupled interaction between AB/rotor and other subsystems presented here, the findings in this paper suggest that AB can be successfully employed into the coupled featured system, but the complicated limit cycle should be also considered as well. Finally, it is hoped that this work will provide valuable insight into use of AB in more practical vibration isolation problem of rotor systems with an additional oscillating mechanical component.



---

*Source: 4083897-2018-06-07.xml*