The Stability of Automatic Ball Balancers (Paper ... - Michael I Friswell

the full nonlinear system [5]. We also consider the transient response of a machine fitted with an ADBmechanism. The transient behavior in linear dynamical systems can be very different to that predicted by a linearstability analysis, where the eigenvalues determine the asymptotic behavior. Often, in stable non-normalsystems, transients may exhibit an undesired dynamic response, such as, a large transient growth (or apparentinstability) preceding a long, highly oscillatory, exponential decay or, alternatively, a rapid collapse beforeeventual asymptotic decay to the stable equilibrium [6]. This behavior is of particular concern when consideringa linearization about an equilibrium of a system with strong nonlinearities. In this case, a large transient growthmay result in a trajectory of the full nonlinear system coming close to regions of state space where thelinearization breaks down. Thus the eigenvalues of the linearization do not always correctly predict the behaviorof the full nonlinear system, particularly in a finite time interval [8]. Hence an analysis and understanding of thesensitivity to perturbation and the ensuing transient response is necessary for the robust design of a practicaldevice.2 EQUATIONS OF MOTIONFigure 1 shows a schematic of an ADB with two balancing masses. Movement is confined to the planardirections X and Y. The equations of motion describing this system are derived using Lagrange’s equation [5].The equations describing the displacement in the X and Y directions are given by2MX + c X + k X −M εψ cos ψ−MεψsinψX Xn∑ i i i i ii=12{ ( ) ( ) ( ) ( )}+ m X −R ψ + φ sin ψ+ φ − R ψ + φ cos ψ+ φ = 0n2MY + cYY + kYY −M εψ sin ψ+ M εψcosψ+ Mg + ∑ migi=1n(2)2+ ∑ mi{ Y + R( ψ+φ i) cos( ψ+φi) −R( ψ+φ i) sin ( ψ+φ i)} = 0i=1where ψ is the angular position of the rotor. Finally, the equation of motion of the ith ball is given by2−mR i ⎡X sin ( ψ+φi) −Y ⎣cos ( ψ+φ i) ⎤⎦+ mR i ( ψ+φ i) + Diφ i + mgR i cos( ψ+φ i)= 0 . (3)These equations can be simplified by assuming equal masses m of the balancing balls, an equal viscous dragforce D acting on each ball, a constant angular velocity of the rotor ω ( ψ =ω t ) , and isotropic suspension of therotor c (damping constant) and k (spring constant). Finally, we neglect the effect of gravity on the motion of theballs, that is we set g=0.Furthermore, we can write the equations in dimensionless form by introducing the following nondimensionalstatesX YX = , Y , t nt,R= R=ω (4)and the dimensionless parametersω m ε c DΩ= , µ= , δ= , ς= , β= .ω22n M R kM mR ωnwhere ω n = k/M is the natural frequency of the rotor.Finally, we obtain a time independent set of equations by transforming to a rotating frame of reference, using( ) ( ) ( ) ( )X = xcos Ωt − ysin Ω t , Y = xsin Ω t + ycosΩ t . (6)This results in the following second-order dimensionless system describing the ADB in the rotating frame:(1)(5)2 Paper-ID 74