An automatic dynamic balancer in a rotating mechanism with time-varying angular velocity
AAn automatic dynamic balancer in a rotating mechanism withtime-varying angular velocity
James A. Wright , Linyu Peng ∗
1. Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK2. Waseda Institute for Advanced Study, Waseda University, Tokyo 169-8050, Japan
Abstract
We consider the system of a two ball automatic dynamic balancer attached to a rotating discwith nonconstant angular velocity. We directly compare the scenario of constant angular velocitywith that when the acceleration of the rotor is taken into consideration. In doing so we showthat there are cases where one must take the acceleration phase into consideration to obtain anaccurate picture of the dynamics. Similarly we identify cases where the acceleration phase of thedisc may be ignored. Finally, we briefly consider nonmonotonic variations of the angular velocity,with a view of maximising the basin of attraction of the desired solution, corresponding to dampedvibrations.
Keywords: attractor; automatic dynamic balancer; basin of attraction; mechanical system;ramped velocity
There is a vast array of mechanical systems in which imbalance causes undesirable vibrations, e.g.washing machines, drills, vacuum cleaners, jet engines, fans, etc. In many instances the imbalance iscaused by imperfections in the manufacturing process, however there are also cases where imbalanceis caused by the operation of the machinery or resulting from sustained damage. One well knownexample in household appliances is washing machines. When a washing machine which is filled withclothes spins, a large amount of imbalance is created due to the uneven distribution of the clothes.This causes the washing machine to vibrate. To reduce these vibrations, washing machines have aheavy concrete weight in the base. This makes them difficult to move, and still does not completelyquell the vibrations. Another example is the danger posed to aircraft by birds. If a bird is suckedthrough a jet engine of an aircraft it can damage the blades of the engine; in most cases leading toimbalance. So as to avoid dangerous vibrations and maintain control of the aircraft, the pilot is forcedto shut down the engine. Even during normal operation, the vibrations resulting from jet-engines maycause weakness in the structure of aircraft.It is clear that the unbalance of rotating parts in machinery is a common cause of undesirablevibrations and is a widespread problem. Such vibrations can cause damage to both the user and themachine itself, as well as create undesired noise pollution. For rotors with a fixed amount of imbalance,it is sufficient to balance the system only once. However, effects such as thermal deformation, materialerosion and those mentioned above can cause the mass distribution to change, as a result the balancingprocedure may have to be repeated. Furthermore, in cases where imbalance is caused as a result ofthe operation of machinery, it may not be possible to balance the system in advance.An automatic dynamic balancer (ADB), is a passively controlled device which requires no externalforces to eliminate the imbalance of rotating mechanisms. An ADB works using two or more weightedballs housed in a race which is filled with a viscous fluid. As the rotor spins the forces on the balls causethem to move so as to eliminate the imbalance in the rotor and hence quell the vibrations. The ADBsystem has already been extensively investigated in a large amount amount of literature by method ofanalysis, numerical simulation and even practical implementation, see for example [10, 11, 12, 13, 17]and the references contained therein. It was shown in [16, 17] that provided the angular velocity of ∗ Corresponding author. E-mail address: [email protected] Also referred to as an automatic ball balancer (ABB). a r X i v : . [ n li n . C D ] J un he system, ω , is greater than the critical (also called natural) angular velocity, ω c , an ADB may besuccessfully utilised to dampen the vibrations caused by imbalance.However, while an ADB can reduce imbalance and minimise vibrations in a system, previousresearch has shown that it may also cause periodic motion far worse than the original vibrations [4],which pose a greater risk of damage to both the user and the machine. Furthermore in [10] it wasshown that large transient time-spans are necessary in order to maximise the probability of achievingbalance with an ADB . Naturally, large transients are undesired, as damage may already be causedbefore balance is attained. For these reasons, ADB devices are not widely implemented in industry.In an attempt to address the prior mentioned problems there has also been research into modifiedADB devices, see for example [15, 18, 19, 21].In previous literature it is assumed that the system rotates with constant angular velocity, ω , seefor example [5, 6, 11, 13, 26]. This is justified by assuming that the balls are held in position usinga clamping mechanism while the rotor accelerates then released once the desired angular velocity isattained. The use of a clamping mechanism was first proposed in [7, 22] and has since become astandard approach when studying ADB systems. This model ignores the effects of any vibrationswhich occur during the spin-up of the rotor, while the balls are clamped in position.In this paper we consider a rotor for which the angular velocity is a nonconstant function of time,that is, the balls are released before the final angular velocity is attained. In particular, we focus ourattention on a linearly varying angular velocity, which increases from an initial value ω to the final,desired angular velocity ω f , given by ω ( t ) = (cid:40) ω + ( ω f − ω ) tT if 0 ≤ t < T ,ω f if t ≥ T , (1)where T is the time over which the rotation speed increases. Recently there has been a growing interestin ODE systems in which a parameter initially depends on time, see for example [1, 2, 3, 8, 23, 24, 25].However, previous work has focused on systems with one-and-a-half degrees of freedom. Here weextend the work in the literature and consider a basic ADB system, which is an ODE system withhigher degrees of freedom that also has vast applications in engineering and industry. Although weshall only consider a basic ADB design, the interested reader may apply the ideas presented here tomodified designs.The paper is organised as follows. In Section 2 we derive the equations of motion for a rotatingdisc with an ADB attached. Whin deriving the equations the angular velocity is assumed to betime-dependent. The equations of motion are transferred to dimensionless form for the purpose ofnumerical implementation, which is conducted in Sections 3 to 5. In Sections 3 to 5 we use numericalmethods to investigate an ADB system with two balls; we will focus our attention on studying theamplitude of the transient vibrations and the percentage of phase space contained inside the basin ofattraction of the damped solution, i.e. the probability of achieving balance.In Section 3 we study the system with constant angular velocity. This is the approach used inprevious literature, which we later compare with the results in Sections 4 and 5, where the angularvelocity is a nonconstant function of time, given by equation (1). In Section 4 we consider the angularvelocity of the disc as an initially increasing function of time, this models the spin-up of the rotor.The balancing balls in the ADB are released once the rotor reaches the final angular velocity, ω f ; thisapproach is similar to that in previous literature and Section 3, however here we are able to studythe effects of rotor acceleration on the dynamics of the system. In Section 5 we consider a model inwhich the balls are released prior to the final angular velocity. We also briefly consider nonmonotonicvariations of ω ( t ) and the resulting effects on the basins of attraction. Finally in Section 6 we giveconcluding comments. The system consists of an eccentric rotating disc together with an ADB, two or more balls free tomove in a race filled with a viscous fluid and positioned at a fixed distance from the centre of rotationof the disc, see Figure 1. All motion is assumed to be confined to the two-dimensional (
X, Y )-plane.Let L = T − V be the Lagrangian with respect to this mechanical system with generalised external We refer here to the probability of achieving balance with random placement of the balls in the race, i.e. the relativesize of the balanced state’s basin of attraction in phase space ( φ , . . . , φ n ), where φ i specifies the location of the i -thball. ψ ( t ) measuresthe angle made between the centre of mass of the disk and the horizontal axis. The coordinates φ i ( t )measure the angle between the i -th ball and the centre of mass of the disk. The radius of the diskis denoted by R and ε is the distance between the centre of mass and the centre of rotation. Theparameters c X and c Y are linear damping constants which act on the rotor in the X and Y directions,respectively, and k X , k Y are linear spring constants.forces Q = ( − c X ˙ X, − c Y ˙ Y , ˜ M , − D i ˙ φ i ), where T and V are the kinetic energy and potential energy,respectively. The equations describing the motion of the system are the Euler-Lagrange equationsdd t (cid:18) ∂L∂ ˙ q k (cid:19) − ∂L∂q k = Q k , (2)where q = ( X, Y, ψ, φ i ) is the generalised coordinate system. The coordinates X ( t ), Y ( t ) measure thedisplacement of the disc due to vibrations. The coordinate ψ ( t ) measures the angle made between thecentre of mass of the disk and the horizontal axis. The coordinates φ i ( t ) measure the angle betweenthe i -th ball and the centre of mass of the disk. The parameters c X and c Y are linear dampingconstants which act on the rotor in the X and Y directions, respectively. The value ˜ M is the momentdriving the system, and D i is the linear drag coefficient associated with the i -th ball. The kineticenergy T and potential energy V are, respectively, given by T = 12 I z ˙ ψ + 12 M (cid:20)(cid:16) ˙ X − ε ˙ ψ sin ψ (cid:17) + (cid:16) ˙ Y + ε ˙ ψ cos ψ (cid:17) (cid:21) + 12 n (cid:88) i =1 m i (cid:20)(cid:16) ˙ X − R (cid:16) ˙ ψ + ˙ φ i (cid:17) sin ( ψ + φ i ) (cid:17) + (cid:16) ˙ Y + R (cid:16) ˙ ψ + ˙ φ i (cid:17) cos ( ψ + φ i ) (cid:17) (cid:21) , and V = 12 k X X + 12 k Y Y + M gY + n (cid:88) i =1 m i g ( Y + R sin( ψ + φ i )) , where M is the mass of the disc (without the balancing balls), m i is the mass of the i -th ball, I z is the moment of inertia of the rotor about the center of rotation of the disc, R is the radius of thedisc, ε is the distance between the centre of mass and the centre of rotation of the disc, g is theacceleration due to gravity, and k X , k Y are linear spring constants acting on the rotor in the X and Y directions, respectively. The equations of motion for an ADB device with constant angular velocityhave been calculated in much of the literature, see for example [4, 11, 13, 14, 20]. We refer the readerin particular to [11] where the equations are derived in detail. The equations of motion (3)-(6) aresimilar to those in the literature except for the addition of terms including ¨ ψ , which disappear when3he angular velocity ω is constant. Note also that ˙ ψ = ω only if ω is constant. M ¨ X − M ε ˙ ψ cos ψ − M ε ¨ ψ sin ψ + n (cid:88) i =1 m i (cid:104) ¨ X − R ( ¨ ψ + ¨ φ i ) sin( ψ + φ i ) − R ( ˙ ψ + ˙ φ i ) cos( ψ + φ i ) (cid:105) + k X X = − c X ˙ X, (3) M ¨ Y − M ε ˙ ψ sin ψ + M ε ¨ ψ cos ψ + n (cid:88) i =1 m i (cid:104) ¨ Y + R ( ¨ ψ + ¨ φ i ) cos( ψ + φ i ) − R ( ˙ ψ + ˙ φ i ) sin( ψ + φ i ) (cid:105) + k Y Y + M g + n (cid:88) i =1 m i g = − c Y ˙ Y , (4) I z ¨ ψ − M ε ¨ X sin ψ + M ε ¨ Y cos ψ + M ε ¨ ψ − n (cid:88) i =1 m i (cid:104) R (cid:16) ¨ X sin( ψ + φ i ) − ¨ Y cos( ψ + φ i ) (cid:17) + R ( ¨ ψ + ¨ φ i ) (cid:105) + n (cid:88) i =1 m i gR cos( ψ + φ i ) = ˜ M , (5) − m i R (cid:104) ¨ X sin( ψ + φ i ) − ¨ Y cos( ψ + φ i ) (cid:105) + m i R ( ¨ ψ + ¨ φ i ) + m i gR cos( ψ + φ i ) = − D i ˙ φ i . (6)Since the disc is driven, the angular velocity of the disc ω ( t ) is controlled. Therefore, the systemis not governed by the torsional load and we may neglect equation (5). The remaining equations ofmotion may be simplified by assuming that all the balls in the balancer have equal mass m and exertequal viscous drag D , i.e. m i = m, D i = D, for i = 1 , , . . . , n. We may non-dimensionalise the equations by setting¯ X = XR , ¯ Y = YR , ¯ t = ω c t, and introducing the dimensionless parameters µ = mM , λ = εR , G = gRω c , where ω c = (cid:112) k/M is the critical (also called natural) angular velocity of the disk. More precisely, ω c is the angular velocity which resonates with the springs, given the mass of the disc; it is also oftenreferred to as the critical (or natural) frequency. This form of ω c assumes isotropic suspension of therotor, namely { c, k } = { c X , k X } = { c Y , k Y } . We also introduce the following dimensionless parameters ζ = c √ kM , β = DmR ω c . The external damping ratio ζ , relates to the damping induced by the springs which are attached tothe outside of the ball race. The internal damping ratio β , represents the amount of drag on eachball created by the viscous fluid in the race. Note that all the parameters are assumed to be positive.This is without loss of generality since negative ε implies an eccentric centre of mass located on thenegative axis, which can be mapped back to the positive axis by a simple change of coordinates. Thedimensionless form of equations (3), (4) and (6) is given by(1 + nµ ) ¨¯ X + ζ ˙¯ X + ¯ X = λ ˙ ψ cos ψ + λ ¨ ψ sin ψ + µ n (cid:88) i =1 (cid:20) ( ¨ ψ + ¨ φ i ) sin( ψ + φ i ) + (cid:16) ˙ ψ + ˙ φ i (cid:17) cos( ψ + φ i ) (cid:21) , (1 + nµ ) ¨¯ Y + ζ ˙¯ Y + ¯ Y = λ ˙ ψ sin ψ − λ ¨ ψ cos ψ − (1 + nµ ) G − µ n (cid:88) i =1 (cid:104) ( ¨ ψ + ¨ φ i ) cos( ψ + φ i ) − (cid:16) ˙ ψ + ˙ φ i (cid:17) sin( ψ + φ i ) (cid:105) , φ i − ¨¯ X sin( ψ + φ i ) + ¨¯ Y cos( ψ + φ i ) + ¨ ψ + G cos( ψ + φ i ) = − β ˙ φ i . Now the dots represent differentiation with respect to the new time ¯ t . It is easily checked that setting ψ (¯ t ) = Ω¯ t , with Ω = ω/ω c constant, the equations of motion in [11] are recovered. For ease we dropthe ‘bar’ notation hereafter. The system may be put into the rotating frame using the transformation X = x cos ψ − y sin ψ, Y = x sin ψ + y cos ψ, which results in the equations(1 + nµ ) (cid:18) ¨ x ¨ y (cid:19) + (cid:18) ζ − nµ ) ˙ ψ nµ ) ˙ ψ ζ (cid:19) (cid:18) ˙ x ˙ y (cid:19) + (cid:32) − (1 + nµ ) ˙ ψ − (cid:16) ζ ˙ ψ + ¨ ψ (cid:17) ζ ˙ ψ + ¨ ψ − (1 + nµ ) ˙ ψ (cid:33) (cid:18) xy (cid:19) = λ (cid:18) ˙ ψ − ¨ ψ (cid:19) − (1 + nµ ) G (cid:18) sin ψ cos ψ (cid:19) + µ n (cid:88) i =1 (cid:16) ˙ ψ + ˙ φ i (cid:17) ¨ ψ + ¨ φ i − (cid:16) ¨ ψ + ¨ φ i (cid:17) (cid:16) ˙ ψ + ˙ φ i (cid:17) (cid:18) cos φ i sin φ i (cid:19) , (7)and¨ φ i + β ˙ φ i + ¨ ψ + G cos( ψ + φ i ) = (cid:16) ¨ x − ψ ˙ y − ˙ ψ x − ¨ ψy (cid:17) sin φ i − (cid:16) ¨ y + 2 ˙ ψ ˙ x − ˙ ψ y + ¨ ψx (cid:17) cos φ i . (8)Taking the angular velocity as defined in (1), we have ψ ( t ) = t Ω( t ) = (cid:40) Ω t + (Ω f − Ω ) t T if 0 ≤ t < T , Ω f t if t ≥ T . (9) Throughout the remainder of the paper we restrict our attention to an ADB device with two balls,using equations (7) and (8) to describe the motions of the system. In all cases the balancing balls arereleased at an angular velocity Ω r > Ω c = 1. As a consequence, the centrifugal forces are much greaterthan the gravitational forces on the balls. Therefore we may ignore the effects of gravity, i.e. we set G = 0. Moreover, gravity can be neglected by assuming the rotor is held in a horizontal position.We begin by considering an ADB device attached to a disc with constant angular velocity, whichwe later compare to a disc which accelerates from a stationary position. We set the initial conditionsas x = y = ˙ x = ˙ y = ˙ φ = ˙ φ = 0 , and study a region R of phase space ( φ , φ ), which is definedby φ , φ ∈ [0 , π ). Throughout, the sizes of basins of attraction are given as the percentage of theregion R which they cover. The estimates of the basin sizes are calculated using either 100 000 or 300000 pseudo-random initial conditions chosen inside the region R ; this gives an error in the basin sizesin either the first or second decimal place with a 95% confidence interval, see Table 1 of [25].For a balanced state to exist, the balls must be heavy enough to counteract the imbalance of thedisc. This yields the condition (cid:88) i =1 Rm i ≥ εM = ⇒ (cid:88) i =1 µ i ≥ λ. Note we have assumed m = m = m and µ = µ = µ . Preliminary simulations have shown that –provided the above condition holds – smaller values of µ result in lower amplitude vibrations duringthe transient phase. However, in industrial applications it is likely that the amount of imbalancegenerated will be not known prior to using the machinery. Thus, the smaller the value of µ , the morelimited the device is with regard to the amount of imbalance it can rectify. Moreover, for reasonssuch as safety of operation, it is unlikely that machinery will operate with λ and µ chosen such thatthe above equation is too close to equality. Therefore we set µ larger than necessary to counter theimbalance of the disc and fix λ = 0 . µ = 0 .
05 in all simulations. In [10] the authors consideredthe effects of the internal and external damping ratios β and ζ , respectively. They showed that, forsmall fixed β , increasing ζ reduced the time-span over which transient dynamics occur. However, they Simply, the disc starts without any initial displacement or vibration and the balancing balls are at rest. ζ drastically reduces the probability of achieving balance; a resultof unwanted vibrations becoming asymptotically stable. Furthermore, the authors commented thattaking ζ too large could lead to the loss of an attracting balanced state altogether. In preliminarysimulations we found that increasing both β and ζ , the transient time-span for the system is shortenedfurther still. Moreover, we find that also increasing β leads to an increase in the basin of attractioncorresponding to the desired balanced state, see Table 1 where ζ = 0 . β = 0 .
01, 0.05, 0.1 and0.25. Another advantage of increasing β is that the risk of ball lag is reduced; this is discussed furtherin Section 5.Table 1: Relative sizes of the basins of attraction corresponding to the balanced state x = y = 0 forconstant Ω. The parameter values are fixed as λ = 0 . µ = 0 .
05 and ζ = 0 .
5, with various values for β . The sizes of the basins of attraction are given as the percentage of the region R which they cover,and are calculated using 300 000 pseudo-random initial conditions.Ω β = 0 . β = 0 . β = 0 . β = 0 . β results in anincrease in the basin of attraction corresponding to the balanced state.While larger β is clearly preferable when implementing an ADB device, the system exhibits morevaried dynamics with respect to Ω when β = 0 .
05. As the variation is preferable when studying theeffects of the acceleration of the disc on the transient and asymptotic dynamics, we shall focus ourattention on the system with β = 0 .
05. For the chosen parameter values, four non-balanced stateattractors exist, which we name V to V . Examples of the non-balanced state attractors are given inFigure 2. Of course, the exact magnitude of the radial vibration for each V i depends on the particularvalue of Ω. The relative sizes of the basins of attraction for various values of Ω are given in Table 2.6 a) (b)(c) (d) Figure 2: Plots of the radial displacement r = (cid:112) x + y for the “non balanced state” attractingsolutions. Figures (a) and (b) show the radial vibrations V V
2, respectively, with Ω = 2 . V V
4, respectively, with Ω = 4 .
4. The parametervalues are fixed as λ = 0 . µ = 0 . ζ = 0 . β = 0 . φ , φ ) for constant Ω. The parametervalues are fixed at λ = 0 . µ = 0 . ζ = 0 . β = 0 .
05. The fixed initial conditions are x = y = ˙ x = ˙ y = ˙ φ = ˙ φ = 0. The sizes of the basins are given by their percentage coverage of theregion R = [0 , π ) × [0 , π ), and are calculated using 300 000 pseudo-random initial conditions.Ω Balanced V V V V × − × − × − Time-dependent angular velocity with Ω r ≡ Ω f We now consider an ADB device attached to a disc which accelerates in a linear manner from aninitial velocity Ω , to a final velocity Ω f , see equation (9). To simulate a real life scenario we setΩ = 0, i.e. the disc accelerates from a stationary position. The time-span over which the discaccelerates is denoted by T . Initially the balancing balls are held in position and are then releasedonce the disc reaches terminal velocity, that is Ω r ≡ Ω f where Ω r is the angular velocity at which theballs are released. This representation more accurately models the imbalance of the disc during theacceleration phase. Fixing Ω f = 2 .
4, we consider various values of T and plot the radial displacement r = (cid:112) x + y against time t , see Figure 3. Note that the balls are located opposite each other so asnot to alter the imbalance of the disc while it accelerates.In Figure 3(f) T = 300: as the disc accelerates, the imbalance causes the displacement of the discto increase. After an initial peak, the displacement settles down to an approximately constant value;this happens before the balls are released. Taking T smaller, the dynamics happen over a shorterinterval of time, see Figures 3(d),(e) where T = 50, 100, respectively. However, if the accelerationoccurs over a very short time interval – see Figures 3(a)-(c) – the system is unable to react. In thisinstance, the balancing balls are released whilst the radial displacement is still increasing. Hence theprecise time at which the balls are released has an affect on the transient vibrations and may alsoalter the asymptotic solution.The sizes of the basins of attraction for several values of T and Ω r = Ω f = 2 . T large enough, the exact value of T does not have any significantaffect on the sizes of the basins of attraction. However, for small T the basin sizes do differ; in thisinstance, with the exception of T = 1, the basin corresponding to the balanced state is slightly larger.This is interesting as it is commonly believed that quickly accelerating up to the final angular velocityallows one to ignore the vibrations caused during the acceleration phase. In fact, the faster the discaccelerates, the further the results are from the constant Ω regime, compare the results in Table 3with those for Ω = 2 . T shown, the sizes of the basins differ from thosein Table 2 with Ω = 2 .
4; in particular the basin of attraction corresponding to the balanced state isapproximately 10% larger.Table 3: Sizes of the basins of attraction with Ω = 0, Ω f = 2 . T . The fixedinitial conditions are x = y = ˙ x = ˙ y = ˙ φ = ˙ φ = 0 and the basins are approximated using 100 000initial conditions in the plane ( φ , φ ). T Balanced V V − − − − r = Ω f = 5 were also conducted. In this instance we found that the basinsof attraction did not vary significantly, regardless of the value of T . In particular, the sizes of thebasins presented in Table 2, in which Ω is constant, provide a good approximation to when Ω( t )increases from Ω = 0. In order to understand the underlying reason, we may think of the system asevolving over two time intervals, namely [0 , T ) and [ T , ∞ ). During the interval [0 , T ), the systemis a transient system , see [8], as the angular velocity of the disc varies with time. The evolution ofinitial conditions during this first time interval set up the “initial conditions” for the steady system in the second time interval, [ T , ∞ ). The “initial conditions” for the steady system will likely not8 a) (b)(c) (d)(e) (f) Figure 3: The radial displacement r = (cid:112) x + y plotted against time. The initial conditions are fixedat x = y = ˙ x = ˙ y = ˙ φ = ˙ φ = 0, φ = − π , φ = π . The parameters are fixed as µ = 0 . ζ = 0 . λ = 0 . β = 0 .
05, Ω = 0, Ω f = 2 .
4. The time-spans are T = 10 − , 10 − , 5, 50, 100 and 300 in(a)-(f), respectively. 9e evenly distributed, but have formed dense clusters in phase space, depending on the attractorswhich exist for the transient system . The location and density of these clusters will be dependent onthe precise evolution of the transient system , for more details see [25, 23]. In particular, if the basinof attraction for the balanced state is large in the steady system , it is likely to capture most of thetrajectories resulting from the transient system . Therefore we may postulate that, the accelerationof the disc has more significant effects on the asymptotic behaviour of the system when the basinsof attraction in the constant angular velocity regime are smaller. However, in order to rigorouslyapply the methods in [25] to understand the changes in the basins of attraction when Ω( t ) varies,one must study the basins of attraction in the full phase space ( x, y, φ , φ , ˙ x, ˙ y, ˙ φ , ˙ φ ) . Althoughpossible, numerically calculating the eight-dimensional basins of attraction would take a vast amountof computation time and the resulting data would be difficult to analyse. Moreover, while one couldnumerically implement the ideas in [25] to predict the asymptotic behaviour of trajectories, there isno obvious way of displaying the basins in eight dimensions, making visual analysis impossible. Itis therefore not beneficial to rigorously implement the ideas of [25] in this case, although it can behelpful to keep them in mind.In Table 4 we fix T = 500 and present results for various values of Ω f . As one may expect (giventhe sizes of the balanced state’s basin of attraction in the constant Ω regime), for Ω f ≥ . t ) varying from Ω = 0 to Ω f over a time-span T = 500. The balls are released at Ω r = Ω f . The parameter values are set to λ = 0 . µ = 0 .
05 and ζ = 0 . β = 0 .
05. The fixed initial conditions are x = y = ˙ x = ˙ y = ˙ φ = ˙ φ = 0 and the basins areapproximated using 100 000 initial conditions in the plane ( φ , φ ).Ω f Balanced V V V V × − Ω r < Ω f We now consider releasing the balancing balls at Ω r < Ω f , which we directly compare with the regimestudied in Section 4, where Ω r = Ω f . That is, we compare the two scenarios:(i) The angular velocity increases linearly from Ω( t ) = Ω ≡ t ) = Ω f over a time T , atwhich point the balls are released, that is Ω r = Ω f .(ii) The angular velocity increases linearly from Ω( t ) = Ω ≡ t ) = Ω f over a time T . Theballs are released at some angular velocity Ω r with Ω c < Ω r < Ω f .It is clear from Table 4 that, in scenario (i) with the chosen parameter values, when Ω f ≥ . φ , φ ). This may also be seen in Table 2, where theacceleration phase of the disc is ignored. We therefore expect to see little or no beneficial change tothe basins of attraction when considering scenario (ii) with Ω f ≥ . With the assumption that the balls are held in position until t = T , one may neglect the ˙ φ and ˙ φ directions. ζ = 0 . β = 0 .
01; compare (i) where Ω r = Ω f = 5with (ii) where the balls are released early, with Ω r = 2. However, if the internal damping β is chosento be sufficiently large, the effects of ball lag can be reduced; since the viscous fluid inside the raceexerts enough drag force on the balls to prevent them from lagging too far behind the acceleration ofthe disc.Figure 4: The top plot (i) shows the vibrations when the balls are released at Ω r = Ω f = 5. Thebottom plot (ii) shows the vibrations for Ω r = 2. The parameters are λ = 0 . µ = 0 . ζ = 0 . β = 0 .
01 and Ω = 0, and we fix the initial conditions x = y = ˙ x = ˙ y = ˙ φ = ˙ φ = 0, φ = − π/ φ = π/ ζ = 0 . β = 0 .
05 – ball lag does pose some risk, see Figure 5(c).When the balls are released at Ω r < Ω f , the transient vibrations whilst the disc accelerates are worsethan those for the system with Ω r = Ω f . That said, releasing the balls early is not without benefit.For instance, when Ω r = 2 . f = 2 .
4, provided the acceleration time T is not too small, thebasin of attraction corresponding the balanced state is larger than those when Ω r = Ω f = 2 .
4. Thismay be seen by comparing the results in Table 5 where Ω r = 2 . r = Ω f = 2 .
4. Depending on the value of T it is possible for the balanced state to attract up to20% more of the region R when the balls are released early. However, releasing the balls early doesnot necessarily result in preferable asymptotic dynamics. For example, in cases where Ω f is large, thebasin of attraction corresponding to the balanced state is also large when Ω r = Ω f , see Table 4. Then,releasing the balls early may result in little or no improvement to the size of the basin of attraction,while still possibly resulting in worse transient dynamics. Furthermore, releasing the balls early canreduce the size of the desired basin of attraction. For example, Table 6 shows results for the systemwith Ω r = 2, 3 and Ω f = 5; in all cases the safe basin is smaller as a result of releasing the balls early,compared with Ω r = Ω f = 5, see Table 4. 11 a) (b)(c) (d) Figure 5: The top plot (i) of each figure shows the vibrations when the balls are released at Ω r =Ω f = 2 .
4. The bottom plot (ii) shows the vibrations for Ω r = 2 .
2. The parameters are λ = 0 . µ = 0 .
05. (a) ζ = 0 . β = 0 .
01. (b) ζ = 0 . β = 0 .
25. (c) ζ = 0 . β = 0 .
05. (d) ζ = 0 . β = 0 . x = y = ˙ x = ˙ y = ˙ φ = ˙ φ = 0, φ = − π/ φ = π/ = 0, Ω r = 2 .
2, Ω f = 2 . T . The parameters are fixed as µ = 0 . ζ = 0 . λ = 0 .
01 and β = 0 .
05. The relative sizes of thebasins were calculated using 100 000 pseudo-random initial conditions inside the region R . T Balanced V V
100 48.34 05.90 45.76200 55.71 05.55 38.74300 72.17 03.99 23.84400 86.61 03.51 09.89500 89.40 03.36 07.23600 90.59 02.94 06.46700 91.83 02.83 05.80800 91.93 02.41 05.6512able 6: Relative sizes of the basin of attraction corresponding to the balanced state x = y = 0. Theparameters are set to µ = 0 . ζ = 0 . λ = 0 .
01 and β = 0 .
05. The results shown are for Ω r = 2, 3,with Ω = 0 and Ω f = 5. The relative sizes of the basins were calculated using 300 000 pseudo-randominitial conditions inside the region R . T Ω r = 2 Ω r = 3100 14.92 49.34150 50.13 77.86200 68.63 84.44300 81.18 89.62400 84.31 93.03500 85.66 94.271000 88.16 96.05Let us now briefly consider a nonmonotonic model for the angular velocity. It may be seen fromTable 4 that, for the chosen parameters, the balanced state has a much larger basin of attraction whenΩ f = 3 than for Ω f = 2. It may be the case that one wishes to set Ω f = 2, without sacrificing sucha large portion of the basin of attraction. This can be achieved by increasing the angular velocity toΩ( t ) = Ω r = 3, maintaining the higher rotation speed for a short period of time then decelerating theangular velocity of the disc to the desired rotation speed Ω f = 2. This model of the angular velocitymay be written as Ω( t ) = Ω + (Ω r − Ω ) tT if 0 ≤ t < T , Ω r if T ≤ t < T , Ω r + (Ω f − Ω r ) t − T T − T if T ≤ t < T , Ω f if t ≥ T , (10)so that the system has four distinct time-intervals [0 , T ), [ T , T ), [ T , T ) and [ T , ∞ ). If the discattains balance at Ω( t ) = Ω r = 3, then provided the system does not suffer with ball lag, or indeed theballs move too fast when the disc decelerates, the disc should remain balanced when Ω( t ) = Ω f = 2,see [23, 24, ? ] for further details. By suitably manipulating the angular velocity and choosing thevalues of T , T and T , one is able to obtain a large basin of attraction for the balanced state withΩ f = 2, see Table 7. In fact, on comparison with Table 4 we find approximately a 30% increase inthe basin’s size, and a 40% increase compared with the predictions of Table 2 where Ω is constant.Table 7: Sizes of the basin of attraction corresponding to the balanced state x = y = 0. Theparameters are fixed as µ = 0 . ζ = 0 . λ = 0 .
01 and β = 0 .
05. The angular velocity is modelledby equation (10) with Ω = 0, Ω r = 3 and Ω f = 2. The relative sizes of the basins were calculatedusing 300 000 pseudo-random initial conditions inside the region R . T T T Basin size50 400 500 98.43100 300 500 97.19100 400 500 97.64200 400 500 93.64
We have compared the dynamics for the system of an ADB attached to a rotating disc with bothconstant and nonconstant angular velocities. For the parameter values investigated we numericallyshowed that provided the final angular velocity is high enough, one may ignore the spin-up phase ofthe rotor and still obtain reasonable estimates to the basin sizes. However, we also noted that thisis likely due to the large size of the basin of attraction corresponding to the balanced solution; if asystem is such that the balanced solution has a small basin of attraction for high angular velocities itmay not be possible to ignore the spin-up phase of the rotor.In cases where the angular velocity is slower, and hence the basin of attraction correspondingto damped vibrations is smaller, the initial acceleration of the disc is important when studying theasymptotic dynamics. Furthermore, accelerating quickly up to the final angular velocity does not13llow the scenario of constant angular velocity to serve as a good approximation to the asymptoticdynamics.Releasing the balancing balls early can result in a higher chance of achieving balance when thesystem operates at lower velocities, i.e. when the desired basin is smaller. However, it can also resultin worse transient dynamics. At the end of Section 5 we briefly considered nonmonotonic variationsof the angular velocity and showed that, by suitable manipulation, one can significantly increase thesize of the desired basin of attraction, corresponding to damped vibrations.In addition we have shown that increasing the internal damping in conjunction with the externaldamping results in shorter transient times and an increase in the size of the basin of attractioncorresponding to the desired solution. This is also beneficial as it allows one to minimise the risk ofundesired vibrations resulting from ball lag. This may be coupled with releasing the balls early toobtain short transient time-spans and a large basin of attraction corresponding to damped vibrations.In mechanical systems it is possible to control the internal damping by increasing/decreasing theviscosity of the liquid in the ball race. Therefore, in industrial applications, it should not be difficultto implement the ideas presented here.This work constitutes the first steps to investigating an ADB device attached to a disc with varyingangular velocity. In the process, we have extended the work already available in the literature which isconcerned with time-dependent parameters in ODE systems with one-and-a-half degrees of freedom,to systems with higher degrees of freedom. It would be interesting to study the system further, inparticular, one could study the basins of attraction in the plane ( λ, φ ) with φ fixed as φ + π . Itwould also be interesting to consider the parameter plane ( λ, µ ) with φ , φ fixed. Such studies wouldallow engineers to predict the amount of eccentricity which could be balanced, given a particular ADBset up. Acknowledgements
LP is partially supported by JSPS Grant-in-Aid for Scientific Research (No. 16KT0024), the MEXT‘Top Global University Project’, Waseda University Grant for Special Research Projects (No. 2019C-179, No. 2019E-036) and Waseda University Grant Program for Promotion of International JointResearch.
References [1] Bartuccelli, M. V., Deane, J. H. B. and Gentile, G. (2012) Attractiveness of periodic orbits in paramet-rically forced systems with time-increasing friction.
J. Math. Phys. , 102703, DOI: 10.1063/1.4757650.[2] Bishop, S. R. and Galvanetto, U. (1993) The influence of ramped forcing on safe basins in a mechanicaloscillator. Dynam. Stabil. Syst. , 73–80.[3] Bishop, S. R. and Galvanetto, U. (1993) The behaviour of nonlinear oscillators subjected to rampedforcing. Meccanica , 249–256.[4] Chung, J. and Ro, D. S. (1999) Dynamic analysis of an automatic dynamic balancer for rotating mech-anisms. J. Sound Vibration , 1035–1056.[5] Chung, J. and Jang, I. (2003) Dynamic response and stability analysis of an automatic ball balancer fora flexible rotor.
J. Sound Vibration , 31–43.[6] Ehyaei, J. and Moghaddam, M. M. (2009) Dynamic response and stability analysis of an unbalancedflexible rotating shaft equipped with n automatic ball-balancers. J. Sound Vibration , 554–571.[7] Ernst, H. (1951) Automatic Precision Balancing.
Machine Design , 107–114.[8] Galvanetto, U. and Magri, L. (2013) On the use of the theory of dynamical systems for transient problems. Nonlinear Dyn. , 373–380.[9] Green, K. (2005) Analysis of an automatic dynamic balancing mechanism for eccentric rotors. In FifthEUROMECH Nonlinear Dynamics Conference (eds van Campen, D. H., Lazurko, M. D. and van denOever W. P. J. M.).[10] Green, K., Champneys, A. R. and Friswell, M. I. (2006) Analysis of the transient response of an automaticdynamic balancer for eccentric rotors.
Int. J. Mech. Sci. , 274–293.
11] Green, K., Champneys, A. R. and Lieven, N. J. (2006) Bifurcation analysis of an automatic dynamicbalancing mechanism for eccentric rotors.
J. Sound Vibration , 861–881.[12] Green, K., Champneys, A. R., Friswell, M. I., et al. (2008) Investigation of a multi-ball, automaticdynamic balancing mechanism for eccentric rotors.
Phil. Trans. R. Soc. A , 705–728.[13] Huang, W. Y. and Chao, C. P. (2002) The application of ball-type balancers for radial vibration reductionof high-speed optic disk drives.
J. Sound Vibration , 415–430.[14] Kim, W., Lee, D. and Chung, J. (2005) Three-dimensional modelling and dynamic analysis of an auto-matic ball balancer in an optical disk drive.
J. Sound Vibration , 547–569.[15] Kim, T. and Na, S. (2013) New automatic ball balancer design to reduce transient-response in rotorsystem.
Mech. Syst. Signal Proc. , 265–275.[16] Lee, J. (1995) An analytical study of self-compensating dynamic balancer with damping fluid and ball. Shock Vib. , 59–67.[17] Lee, J. and van Moorhem, W. K. (1996) Analytical and experimental analysis of a self-compensatingdynamic balancer in a rotating mechanism. ASME J. Dyn. Sys., Meas., Control , 468–475.[18] Rezaee, M. and Fathi, R. (2015) Improving the working performance of automatic ball balancer bymodifying its mechanism.
J. Sound Vibration , 375–391.[19] Rezaee, M. and Fathi, R. (2015) A new design for automatic ball balancer to improve its performance.
Mech. Mach. Theory , 165–176.[20] Rodrigues, D. J., Champneys, A. R., Friswell, M. I., et al. (2008) Automatic two-plane balancing forrigid rotors. Int. J. Nonlinear Mech. , 527–541.[21] Sung, C. K., Chan, T. C., Chao, P. C. P., et al. (2013) Influence of external excitations on ball positioningof an automatic balancer. Mech. Mach. Theory , 115–126.[22] Thearle, E. (1932) A new type of dynamic-balancing machine. Trans. ASME , 131–141.[23] Wright, J. A. (2016) Safe basins for a nonlinear oscillator with ramped forcing. Proc. R. Soc. A ,DOI: 10.1098/rspa.2016.0190.[24] Wright, J. A., Bartuccelli, M. and Gentile, G. (2014) The effects of time-dependent dissipation on thebasins of attraction for the pendulum with oscillating support.
Nonlinear Dyn. , 1377–1409.[25] Wright, J. A., Deane, J. H. B., Bartuccelli, M., et al. (2015) Basins of attraction in forced systems withtime-varying dissipation. Commun. Nonlinear Sci. Numer. Simulat. , 72–87.[26] Yang, Q., Ong, E. H. and Sun, J. (2005) Study on the influence of friction in an automatic ball balancingsystem. J. Sound Vibration , 73–99., 73–99.