3.4 Stability Analysis
3.4.3 Stability Analysis for Zero Load Flow
−2
−1 0
Υ↑
Υ↑
unstable Overlap
PS− P∗
X∗
2 4
◦
Figure 3.5: Stability map visualizing stability conditions (3.23), (3.25), (3.27).
As pointed out above, the rectangular notch is unstable for the majority of pressure dif-ferences – practically, its stable region is negligibly small as the corresponding vertical stability boundary is in close proximity of PS − P∗ = 0irrespective of variations in Υ.
The map also reveals the qualitative superiority of a triangular notch over a circular notch. As Υ is increased, the stable regions increase for all geometries.
3.4.3 Stability Analysis for Zero Load Flow
For an ideal valve and zero load flow (ΓOR = 0), the equilibrium position of the valve will be set valued and lie within the interval X ∈ [−1, 1] since the flow function has infinetely many zeros: any point X∗ ∈ [−1, 1] is a zero of the flow function and thus an equilibrium position. This is a feature of the dead region resulting from perfect impermeability when the valve is nominally closed. Equilibria for the resulting (linear) ordinary differential equation lie in the dead region of the ideal valve and are indifferent as the Jacobian
J(X∗) =
0 0 −A2
0 0 1
A1 −ω2 −2Dω
(3.28)
for this situation has the characteristic equation λ3+ 2Dωλ2+ A1A2+ ω2
λ = 0 (3.29)
and the corresponding eigenvalues
λ1 = 0 , (3.30)
λ2 =−Dω + jp
ω2(1− D2) + A1A2, (3.31)
λ3 =−Dω − jp
ω2(1− D2) + A1A2, (3.32)
so that the maximum eigenvalue real part is equal to zero, implying an indifferent equi-librium position.
As discussed in chapter 2, ideal control valves cannot be provided by real-word man-ufacturing technology. In practice, tolerance errors will always generate leakage flow within the valve. It may therefore be asked in how far leakage affects the stability prop-erties of the valve for zero load flow. For an ideal valve, the equilibrium position is set-valued with any point within the dead region being a possible equilibrium point.
Allowing for leakage, the system obtains a unique equilibrium.
It is assumed that leakage flows at two points in the system, namely from pressure supply into the system and from the system into the tank. After non-dimensionalizing, leakage flow for i = {2, 4,
◦
} and a valve spool position within the dead band readsQ = QLi(X, P )
=− qL1i(−X − 1)LLLi(−X − 1) +
q
(PS− P∗− P )qL2i(−X − 1) + q2L1i(−X − 1)L2LLi(−X − 1) + qL1i(X − 1)LLLi(X − 1)
− q
(P∗+ P )qL2i(X − 1) + qL1i2 (X − 1)L2LLi(X − 1) (3.33) with the non-dimensional functions
qL1i(X) = 6T uˆηF
Chpˆ
bLLi(uX)
λLD ΞγF2, qL2i(X) = T2
Ch2pˆγF2b2LLi(uX)λ2LD2. (3.34) In the above functions, the non-dimensional geometric leakage parameter λL = ∆r/D was introduced and viscosity was scaled by ˆηF = 1Passo that it can be represented by Ξ = ηF/ˆηF. The new quantities LLLi(X− 1), LLLi(−X − 1) are the non-dimensionalized channel lengths for the laminar components of leakage for flow into the tank and from pressure supply into the system, respectively, as computed by equation (2.1) et sqq. for different notch geometries i = {2, 4,
◦
}. With this representation of volume flow, the equilibrium position can be computed upon which the Jacobian of the system about this equilibrium may be evaluated. Parameters used for simulation are listed in Table 3.1.One natural choice of parameters by which to vary the system properties and to com-pute stability maps are supply pressure PS and leakage parameter λL. For these, a
Table 3.1: Parameters of the pressure control valve.
Parameter Symbol Value Unit
Valve mass m 0.015 kg
Damping coefficient d 30 Ns/m
Spring stiffness k 1000 N/m
Force components F0, F1 135, 0–1 N
Excitation frequency Ω 240π rad/s
System capacitance Ch 10−11 m3/bar
Piston area A π/4× 0.012 m2
Supplied pressure pS 20− 50 bar
Flow coefficient γF 0.029 p
m3/kg
Valve overlap u 0.0001– 0.0015 m
Diameter of piston bore D 0.01 m
Spool circumference bReg Dπ m
Triangular notch width b4 0.002 m
Triangular notch length a4 0.002 m
Circular notch radius r◦ 0.001 m
Gap height ∆r 15× 10−6 m
Fluid viscosity ηF 0.005 Pas
corresponding dimensional range of 20 − 80bar and 1 × 10−6− 5 × 10−5mwere chosen, respectively. While different system parameters do of course influence the precise lo-cation of the equilibrium position within −1 ≤ X∗ ≤ 1, a numerical analysis reveals that for the majority of parameter constellations investigated the partial derivatives of the volume flow from (3.33) are such that a destabilization of the equilibrium position within −1 ≤ X∗ ≤ 1 does not occur. In these cases, equilibria within −1 ≤ X∗ ≤ 1 are stable irrespective of the notch geometries.
This is most notably due to the overlap u which for this type of valves typically lies in a range of up to 1.5mm. Because decreasing LLLi(−X − 1) implies increasing LLLi(X − 1)(and vice versa), from a certain point onwards in either displacement direction the turbulent component of leakage outflow dominates over its laminar inflow counterpart (and vice versa) as long as valve overlap u is sufficiently large.
For comparatively small overlaps (e.g. with u = 0.5 × 10−3m), however, destabilization through leakage is possible, albeit in a parameter region rarely encountered in prac-tice, i.e. with a very large pressure difference between supply pressure and operating pressure. The mechanism behind this instability is visualized in Figure 3.6.
For increasing supply pressures, leakage into the system increases up to the point where leakage outflow can not compensate leakage inflow anymore: increasing supply pres-sures shift the equilibrium position towards an open control edge in direction of the tank. As a consequence, the valve begins to open nominally in order to achieve zero net
3.20 3.35 3.50 1
1.05
−0.04 0 0.04
PS
X
X0
(a) Equilibrium path and limit cycle evolu-tion for increasing supply pressure PS.
1 1.01 1.02 1.03 1.04
−0.02 0 0.02
X
X0
(b) View of the X, X0
-plane.
Figure 3.6: Birth of a stable limit cycle from supply pressure increase as bifurcation pa-rameter. Green dots represent stable, red points unstable equilibria.
fluid flow within the capacitance and to maintain set-pressure. As outlined above and expressed through stability condition (3.23), for a rectangular notch, an open control edge usually yields unstable behavior with limit cycles occurring about the unstable equilibrium. This can well be seen in Figure 3.6b: the equilibria become unstable once they lie in X∗ > 1 + εi, where εi > 0 is a small non-dimensional quantity depending on the leakage characteristic of the geometries i = {2, 4,
◦
} at hand. The stable limit cycles about the unstable equilibria therefore essentially are relaxation oscillations. The reason some equilibria remain stable for 1 ≤ X∗ ≤ 1 + εi (i.e. within the very prox-imity of X = 1) is that the flow function gradient blends into an instability-generating gradient dQ/dX here as visualized in Figure 2.4b.In Figure 3.7, a corresponding stability map for a rectangular notch for the variation of non-dimensional supply pressure PS and non-dimensional leakage gap height λLis shown.
It can well be seen that larger gap heights require lower supply pressures for the equilib-rium to remain stable. Increasing viscosity increases the stable region – this is intuitive since increasing viscosity decreases total leakage flow, thereby eventually leading to a situation where nominal valve opening is not required anymore to compensate leakage inflow.
For notch geometries other than the rectangular notch, destabilization through leakage is not observed in this parameter range. While it is possible that equilibria are shifted into a region where X∗ > 1 holds, stability conditions (3.25) and (3.27) provide the explanation for the persistence of equilibrium stability in spite of a nominally open control edge.
Destabilization effects from leakage within realistic pressure ranges are therefore to be expected mainly in almost critically lapped valves and operating scenarios involving excessively large pressure differences between supply and operating pressure. In
chap-0.1 2.55 5
·10−3 1.45
3.05 4.65
stable
λL
PS
Ξ = 0.004 Ξ = 0.006 Ξ = 0.008
Figure 3.7: Stability map for a rectangular notch.
ter 6 destabilizing effects of leakage in the context of an almost critically lapped valve will be considered further.