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

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ε( A) { z ( A E): E }Λ = ∈Λ + ≤εwhere Λ is the spectrum (set of eigenvalues) of A+E.Figure 4 shows the ε-pseudospectra of the balanced steady state computed using the Matlab package EIG-TOOL [16]. Parameters were fixed at µ = 0.05, δ=ς = β = 0.01 , and from Fig. 4(a) to (d) the rotation speedΩ= 2.5, 3.0, 3.5, 4.0 , respectively. From outermost to innermost, the contours shown in each panel of Fig. 40 1correspond to ε-pseudospectra for ε= 10 , 10 − 2, 10 − 3, and 10 − . These contours are centered around theeigenvalues, indicated by a black dot. Note that the linearized system is eight-dimensional, the additional foureigenvalues and corresponding pseudospectra are symmetric counterparts of those shown for I ( z) > 0. Thesmallest value of ε for which a contour touches the imaginary axis is also known as the stability radius of thelinearized system. From Fig. 4(a) to (d) the stability radii are given by ε= 1.32075× 10 −3 , ε= 1.64948× 10 −3 ,ε= 1.40770× 10 −3 , and ε= 1.076729× 10 −3 . The first three of these correspond to pseudospectra contourssurrounding the eigenvalue closest to the real axis. However, in Fig. 4(d) the stability radius is given by thepseudospectra contour surrounding the third eigenvalue from the real axis. In other words, as the rotation speedΩ is increased, the eigenvalue most sensitive to perturbation changes from being the one with least imaginarypart to being one with a large imaginary part. Furthermore, the order of the sensitivity of the other eigenvaluesalso changes [6]. This change in sensitivity affects the transient dynamics of trajectories of the system as thesteady state is approached.I(z)0(a)I(z)0(b)−2−2−4−4−6−6−1 0 R(z) 1−1 0 R(z) 1I(z)0−2(c)0(d)I(z)−2−4−4−6−6−1 0 R(z) 1−1 0 R(z) 1Figure 4: Pseudospectra of the balanced steady state of the linearized system describing the ADB; see text forfull details.6 Paper-ID 74
Figure 5 shows transient dynamics of the norm of the matrix exponentialAte of the linearized system A(around the balanced steady state solution) for parameter values corresponding to those used in Fig. 4. In eachpanel a large transient growth is observed which decays to the stable, balanced steady state solution. Thepseudospectra of Fig. 4 give us a lower bound on the size of this transient growth. Specifically, if an ε-pseudospectra contour stretches a distance η into the right-half complex plane, a lower bound on the size of thetransient growth is given by η/ε . In other words, as the system becomes more non-normal, one observes alarger transient growth. This is evident in Fig. 5 where an increase in rotation speed Ω (which increases the nonnormalityof the system) is shown to lead to an increase in the initial transient growth. Furthermore, fromFig. 5(a) to (d) the transient is seen to undergo oscillatory decay at differing frequencies. The frequency of thisdecay is directly related to the sensitivity of the eigenvalues, identified above. Specifically, as the rotation speed20‖e At ‖(a)1000 250 500 750 t 100020‖e At ‖(b)1000 250 500 750 t 100020‖e At ‖(c)1000 250 500 750 t 100020‖e At ‖(d)1000 250 500 750 t 1000Figure 5: Transient behaviour of the norm of the matrix exponential of the linearized system describing theADB.7 Paper-ID 74
Page 1 and 2: 7th IFToMM-Conference on Rotor Dyna
Page 3 and 4: ( n )⎛1+ nµ 0 ⎞⎛x⎞ ⎛ 2ς
Page 8 and 9: Ω is increased the eigenvalues mos

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

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