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where (4.3) can be integrated with the initial conditions of (3.5) to yield for t0 ≤ t ≤ t1, which with (4.9) implies 0 ≤ x(t1) − x(t) ≤ 101000t 1 t y(t)−λx(t)=(y−λx)·e−ε t A(s)ds fort≥t. |y0 − λ1x0| (4.10) (4.5) We now show that A(t) is uniformly positive-valued for t ≥ t0. First note that the multiplicative factor y − λ2x on the rightsideof(4.4)hasthevaluey∗−λ2x∗ =(λ1−λ2)·x∗ at the limiting terminal point (x∗, y∗) of (3.17), and this value is positive because λ1 > λ2 and x∗ > 0. Similarly this fac- tor has a positive value at the initial point (x0, y0) since (1.4) implies y − λ2x ≥ y − λ1x on the wedge (1.9), and the last quantity is positive at the initial point by (3.6). In fact the fac- tor y − λ2 x is uniformly positive along the entire portion 0∗ of the hyperbola (3.7)–(3.8) between the initial state (x0, y0) and the limiting state (x∗, y∗) since this (compact) portion 0∗ of the hyperbola is bounded away from the line y = λ2x; cf. Figure 2. Hence there holds with min[y − λ2x] = min 0∗ x 0 ≤ x ≤ x ∗ [Q(x) − λ2x] > 0, (4.6) Passing to the limit t1 → ∞ in (4.11) and using (3.22) yields the stated estimate of (4.2) for x(t) with constant ξ = μ/κ. A corresponding estimate for y(t) (as in (4.2)) follows directly from the estimate (4.9) for y − λ1 x and the estimate of (4.2) for x, using y = (y − λ1x) + λ1x. Remaining details are omitted. The estimates of (4.2) imply that, for small ε > 0, the functions x(t) and y(t) tend rapidly toward their constant lim- iting values x∗ and y∗. For example (4.2) implies for x upon integration where Q(x) is the function given by the positive root y = Q(x) > 0 of the quadratic energy equation (3.7) considered as a function of y (as in (3.9)). For the other factor on the right side of (4.4), a routine estimation gives λ1 − β(α − γ )y + 2αγ x 2 j + 2β2 y + β(γ − α)x ≥ 2jλ1 +(α+γ)(βy−αx) 2 j + 2β2 y + β(γ − α)x (4.7) ∗ μ t1 −κ·(s−t )/ε e 0 ds ε μ:=max|y−λ2x|= max |Q(x)−λ2x| 0∗ x 0 ≤ x ≤ x ∗ where 0 is the segment of the hyperbola (3.7)–(3.8) extend- ing between (x0, y0) and (x∗, y∗). The integral on the right side of (4.10) can be evaluated, and (4.10) implies 0 ≤ x(t1) − x(t) (4.11) μ −κ·(t−t )/ε −κ·(t −t )/ε ≤κ|y0−λ1x0|e 0 −e 1 0 ≤μ|y −λx|·e−κ·(t−t0)/ε fort >t≥t. κ01010 ∞ |x∗−x(t)|dt≤ ξ·|y0−λ1x0| ε, everywhere in the wedge region (1.9). The expression on the right side of (4.7) is uniformly bounded (cf. (3.15)) and uni- formly positive everywhere on the (compact) portion 0∗ of the energy curve (3.7)–(3.8), including at the endpoints (x0, y0) and (x∗, y∗). These remarks with (4.4) imply the existence of a fixed positive constant κ for which there holds A(t)≥κ>0 for t≥t0 (4.8) where κ will generally depend on the parameters x0, y0, α, β, γ, λ1 and λ2, but κ does not depend on ε. From (4.5) and (4.8) there follows immediately (see the simpler calculation in Section 4 of SK [1998]) 0 < y(t) − λ1x(t) ≤ [y0 − λ1x0] · e−κ·(t−t0)/ε for t ≥ t0. (4.9) The angular momentum equation (3.4) can be integrated between t and t1 > t to yield 1 t1 and an analogous result holds for y(t). Hence for t ≥ t0 the area between the graphs of x ∗ and x (t ) (and similarly, the area between y∗ and y(t)) is small, of order ε, as indicated in Figure 3 for x (t ). Quantitative information can be obtained on the magnitudes of the positive constants κ, ξ and η in (4.2) in terms of the data, along the lines of the result (4.3) of SK [1998] for an analogous parameter κ in the case of the radial inflow turbine. x∗ − x(t) ✻ t0 κ x∗ − x0 • t0 t0+ε t Figure 3 ✲ x(t1)−x(t)=ε y(s)−λ1x(s) y(s)−λ2x(s)ds≥0 t 7PDF Image | Transient Characteristics of Radial Outflow Turbine Generators
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