24.3 Probability-Based Tuning

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The noted contribution of this method is its introduction of the likelihood of success as a criterion in the

search for the blade adjustments (Wang et al., 2005). This method speculates the effectiveness of various

adjustment sets in reducing the vibration and selects the set with the maximum probability of producing

acceptable vibration (see Chapter 5 for useful concepts of random or stochastic vibration). The concept

of this method is explained in the context of a simple example. If the measured vibration from the current

SELECTION

NET

CONDENSED

SIMULATION

NET

TRACK

NET

SELECTION

PACKAGE

VIBRATION

NET

DESIRED

Δ VIBS.

COND.

MODS.

(6)

EXPANDED

BLADE

MODS.

(12)

Δ TRACK

(24)

POSSIBLE

FEEDBACK

PREDICTED

Δ VIBS.

(24)

(12)

(24) MODS.

PREDICTED

Δ VIBS.

(24)

FIGURE 24.3 Schematic of the rotor tuning system. The numbers inside parentheses represent the number of

inputs or outputs of individual nets.

Helicopter Rotor Tuning 24-5

© 2005 by Taylor & Francis Group, LLC

flight is denoted by Vjðk 2 1Þ and the estimated

vibration change according to the model is

represented by DV^ jðkÞ ¼ f ðDxÞ as a function of

the blade adjustments, Dx; then the predicted

vibration of the next flight, V^ jðkÞ; can be defined as

V^ jðkÞ ¼ Vjðk 2 1Þ þ DV^ jðkÞ ð24:6Þ

VjðkÞ ¼ V^ jðkÞ þ e^jðkÞ ð24:7Þ

where VjðkÞ denotes the measured vibration for the

next flight. In rotor tuning, the adjustments are

selected according to the predicted vibration,

V^ jðkÞ; whereas the objective is defined in terms of

the measured vibration. The inclusion of the

probability model here is to account for the

inevitable uncertainty in the actual position of

the measured vibration. According to Equation 24.7, the mean value of the measured vibration is equal to

the value of the predicted vibration plus the mean value of the prediction error. However, since the

predicted vibration is a deterministic entity, the probability distribution of the measured vibration is the

same as that of the prediction error. Accordingly, whereas the nominal value of the measured vibration

can be controlled by the blade adjustment, its optimal position within the specification region should be

determined according to its probability distribution. For a case where the prediction error, e^jðkÞ; is zeromean

and normally distributed, as illustrated in Figure 24.4, placing the predicted vibration at the center

of the specification range will be synonymous with maximizing the probability that the measured

vibration will be within the range. The likelihood of success of blade adjustments can therefore be

measured by the area under the probability density function of prediction error located within the

specification region. The blade adjustment set that produces the highest likelihood will be the preferred

adjustment.

The main difficulty with rotor tuning, however, is the limited number of DoFs, which precludes perfect

positioning of the predicted vibration. This point is illustrated in Figure 24.5 for a case where two

vibration components are to be positioned at the center of the specification region with only one

adjustment. If one assumes that the effect of adjustment, Dx; on the change in the two vibration

components, DV^ jðkÞ; can be represented by a linear

model, as

DV^ jðkÞ ¼ aijDx

then the position of the predicted vibration

components will be constrained to the line L in

Figure 24.5. As illustrated in this figure, since it will

be impossible to place the predicted vibration

components at the center, a compromised position

needs to be selected. In this method, the best

compromised position for the predicted vibration

is that which renders the largest probability of

satisfying the specifications for the measured

vibration. This position, for the two-component

vibration example, is one that maximizes

Pr½ðV1; V2Þ[ S􀀉¼

Ð

ðV1 ;V2 Þ[S pðV1; V2ÞdV1 dV2: The

above formulation indicates that the placement

of the predicted vibration requires knowledge of

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Vibration Magnitude

Probability Density

Specification Region

FIGURE 24.4 Illustration of improved placement of

the predicted vibration within the specification range.

−s s

V

^

1

V

^

2

P

L

Q

FIGURE 24.5 Restricted placement of vibration components

within the specification region for a twodimensional

case.

24-6 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

the joint probability density function, pðV1; V2Þ; of the vibration components. In the ideal case of

independent vibration components with equal probability distributions, the loci of the points with equal

probabilities Pr½ðV1; …; VnÞ[ S􀀉 are surfaces of hyperspheres. Such ideal loci for the two-component

vibration example of Figure 24.5 are circles centered at the origin (see Figure 24.5), which lead to

point P as the best compromised position closest on line L to the center of the specification circle.

Point P, however, does not represent the best position if the two vibration components are

dependent or have unequal distributions. The loci of equal probabilities for this more general case are

elliptical, as also shown in Figure 24.5, indicating point Q as the best position on line L for

placing the predicted vibration. The inadequacy of the DoFs illustrated here is exacerbated in rotor

tuning, where 24 correlated vibration components need to be positioned within the specification region

using only four condensed blade adjustments. For the 24-component vector of measured vibration

VðkÞ¼½Vc1ðkÞ; Vs1ðkÞ; …; Vc12ðkÞ; Vs12ðkÞ􀀉T; where Vc and Vs represent the cosine and sine components of

each vibration measurement, respectively, the joint probability density function of measured vibration

for the kth flight, V(k), can be characterized as an N-dimensional Gaussian function:

pðVðkÞÞ¼

1

ð2pÞN=2lFl1=2 exp 2

1

2

e^ðkÞTF21e^ðkÞ

􀀒 􀀓

ð24:8Þ

e^ðkÞ¼VðkÞ2Vðk21Þ2CDxðkÞ ð24:9Þ

where F represents the covariance matrix of the prediction error. Now, if G ¼{lVjl ¼

ffiffiffiffiffiffiffiffiffiffi

V 2

cj þV 2

sj

q

# a; j ¼

1; …; 12} denotes the specification region in 24-dimensional Euclidean space, the blade adjustments, Dxp;

can be selected such that the probability that the measured vibration is within the acceptable range is

maximized (see also Table 24.2). Formally,

Dxp ¼argDx max PrðVðkÞ[ G Þ¼

ð

G

pðVðkÞÞdVðkÞ

􀀒 􀀓

ð24:10Þ

TABLE 24.2 Summary of Probability-Based Tuning

For the input vector:

Dx ¼ ½Dx1; Dx2; Dx3; Dx4􀀉T

where Dx1 and Dx3 denote the combined trim tab adjustments to blade combinations one to three and two to four,

respectively, and Dx2 and Dx4 represent the combined pitch control rod adjustments to blade combinations one to three and

two to four, respectively, the blade adjustments, Dx; can be selected such that the probability that the measured vibration is

within the acceptable range is maximized. Formally,

Dxp ¼ argDx max

􀀑

PrðVðkÞ [ G Þ ¼

Ð

G pðVðkÞÞdVðkÞ

􀀜

where PrðVðkÞÞ denotes the probability of the measured vibration, G denotes the specification region in 24-dimensional

Euclidean space, and pðVðkÞÞ represents the joint probability density of the measured vibration for the kth flight characterized

as an N-dimensional Gaussian function:

pðVðkÞÞ ¼

1

ð2pÞN=2 lFl1=2 exp

􀀑

2

1

2

e^ðkÞTF21e^ðkÞ

􀀜

with

e^ðkÞ ¼ VðkÞ 2 Vðk 2 1Þ 2 CDxðkÞ

representing the predicted error in vibration.

Helicopter Rotor Tuning 24-7

© 2005 by Taylor & Francis Group, LLC

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