Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 14 pptx

3.3.1.5 Labelled Interval Calculus Interval calculus is a method for constraint propagation whereby, instead of des-ignating single values, information about sets of values is propagate

Trang 1

Determination of an optimum conceptual design is carried out as follows:

a) A performance parameter profile index (PPI) is calculated for each performance parameter x i This constitutes an analysis of the rows of the matrix, in which

PPI= n

n

∑

j=1

1

c i j

−1

(3.80)

where n is the number of design alternatives.

b) Similarly, a design alternative performance index (API) is calculated for each design alternative y j This constitutes an analysis of the columns of the matrix,

in which

API= m

m

∑

i=1

1

c i j

−1

(3.81)

where m is the number of performance parameters.

c) An overall performance index (OPI) is then calculated as

OPI=100

mn

m

∑

i=1

n

∑

j=1 (PPI)(API)

(3.82)

where m is the number of performance parameters, n is the number of design

alternatives, and OPI lies in the range 0–100 and can thus be indicated as a per-centage value

d) Optimisation is then carried out iteratively to maximise the overall performance index.

3.3.1.5 Labelled Interval Calculus

Interval calculus is a method for constraint propagation whereby, instead of

des-ignating single values, information about sets of values is propagated Constraint propagation of intervals is comprehensively dealt with by Moore (1979) and Davis (1987) However, this standard notion of interval constraint propagation is not suf-ficient for even simple design problems, which require expanding the interval con-straint propagation concept into a new formalism termed “labelled interval calculus” (Boettner et al 1992)

Descriptions of conceptual as well as preliminary design represent sets of systems

or assemblies interacting under sets of operating conditions Descriptions of detail designs represent sets of components functioning under sets of operating conditions The labelled interval calculus (LIC) formalises a system for reasoning about sets.

LIC defines a number of operatives on intervals and equations, some of which can

be thought of as inverses to the usual notion of interval propagation by the question

‘what do the intervals mean?’ or, more precisely, ‘what kinds of relationships are

Trang 2

possible between a set of values, a variable, and a set of systems or components, each subject to a set of operating conditions?’ The usual notion of an interval constraint is supplemented by the use of labels to indicate relationships between the interval and

a set of inferences in the design context LIC is a fundamental step to understanding fuzzy sets and possibility theory, which will be considered later in detail

a) Constraint Labels

A constraint label describes how a variable is constrained with respect to a given

interval of values The constraint label describes what is known about the values that a variable of a system, assembly, or its components can have under a single set

of operating conditions

There are four constraint labels: only, every, some and none The best approach

to understanding the application of these four constraint labels is to give sample de-scriptions of the values that a particular operating variable would have under a par-ticular set of operating conditions, such as a simple example of a pump assembly that operates under normal operating conditions at pressures ranging from 1,000 to

10,000 kPa.

Only:

< only p 1000, 10000 > means that the pressure, under the specified operating

conditions, takes values only in the interval between 1,000 and 10,000 kPa Pressure does not take any values outside this interval

Every:

< every p 1000, 10000 > means that the pressure, under the specified operating

conditions, takes every value in the interval 1,000 to 10,000 kPa Pressure may or may not take values outside the given interval

Some:

< some p 1000, 10000 > means that the pressure, under the specified operating

con-ditions, takes at least one of the values in the interval 1,000 to 10,000 kPa Pressure may or may not take values outside the given interval

None:

< none p 1000, 10000 > means that the pressure, under the specified operating

conditions, never takes any of the values in the interval 1,000 to 10,000 kPa

b) Set Labels

A set label consolidates information about the variable values for the entire set of systems or components under consideration There are two set labels, all-parts and some-part.

Trang 3

All-parts means the constraint interval is true for every system or component in each selectable subset of the set of systems under consideration For example, in the case

of a series of pumps,

< All-parts only pressure 0, 10000 >

Every pump in the selected subset of the set of systems under consideration oper-ates only under pressures between 0 and 10,000 kPa under the specified operating conditions

Some-part:

Some-part means the constraint interval is true for at least some system, assembly

or component in each selectable subset of the set of systems under consideration

< Some-part every pressure 0, 10000 >

At least one pump in the selected subset of the set of systems under consideration operates only under pressures between 0 and 10,000 kPa under the specified operat-ing conditions

c) Labelled Interval Inferences

A method (labelled intervals) is defined for describing sets of systems or equipment being considered for a design, as well as the operatives that can be applied to these intervals These labelled intervals and operatives can now be used to create inference rules that draw conclusions about the sets of systems under consideration There are five types of inferences in the labelled interval calculus (Moore 1979):

• Abstraction rules

• Elimination conditions

• Redundancy conditions

• Translation rule

• Propagation rules

Based on the specifications and connections defined in the conceptual and pre-liminary design phases, these five labelled interval inferences can be used to reach

certain conclusions about the integrity of engineering design

Abstraction Rules

Abstraction rules are applied to labelled intervals to create subset labelled intervals for selectable items These subset descriptions can then be used to reason about the design

Trang 4

There are three abstraction rules:

Abstraction rule 1:

(only X i )(As,i,S i ) → (only x min

i xl,imax

i xh,i)(A ∩ i S i) Abstraction rule 2:

(every X i )(As,i,S i ) → (every x max

i xl,i min

i xh,i)(A ∩ i S i) Abstraction rule 3:

(some X i )(As,i,S i ) → (some x min

i xl,i max

i xh,i)(A ∩ i S i) where

X = variable or operative interval

i = index over the subset

A = set of selectable items

As,i = ith selectable subset within set of selectable items

S i = set of states under which the ith subset operates

x = variable or operative

xl,i = lowest x in interval X of the ith selectable subset

minixl,i = the minimum lowest value of x over all subsets i

maxi xl,i = the maximum lowest value of x over all subsets i

xh,i = highest x in interval X of the ith selectable subset

mini xh,i = the minimum highest value of x over all subsets i

maxi xh,i= the maximum highest value of x over all subsets i

∩ i S i = intersection over all i subsets of the set of states.

Again, the best approach to understanding the application of labelled interval infer-ences for describing sets of systems, assemblies or components being considered for engineering design is to give sample descriptions of the labelled intervals and their computations

Description of Example

In the conceptual design of a typical engineering process, most sets of systems in-clude a single process vessel that is served by a subset of three centrifugal pumps in parallel Any two of the pumps are continually operational while the third functions

as a standby unit A basic design problem is the sizing and utilisation of the pumps

in order to determine an optimal solution set with respect to various different sets

of performance intervals for the pumps The system therefore includes a subset of three centrifugal pumps in parallel, any two of which are continually operational while one is in reserve, with each pump having the following required pressure rat-ings:

Trang 5

Pressure ratings:

Pump Min pressure Max pressure

Labelled intervals:

X1= < all-parts every kPa 1000 10000 > (normal)

where

xl,1 = 1,000

xl,2 = 1,000

xl,3 = 2,000

xh,1= 10,000

xh,2= 10,000

xh,3= 15,000

Computation: abstraction rule 2:

(every Xi)(As,i , Si) → (every x max i xl,i minixh,i)(A ∩ i Si)

maxi xl,i = 2,000

mini xh,i = 10,000

Subset interval:

< all-parts every kPa 2000 10000 > (normal)

Description:

Under normal conditions, all the pumps in the subset must be able to operate un-der every value of the interval between 2,000 and 10,000 kPa The subset interval

value must be contained within all of the selectable items’ interval values

Elimination Conditions

Elimination conditions determine those items that do not meet given specifications

In order for these conditions to apply, at least one interval must have an all-parts la-bel, and the state sets must intersect Each specification is formatted such that there are two labelled intervals and a condition One labelled interval describes a vari-able for system requirements, while the other labelled interval describes the same variable of a selectable subset or individual item in the subset

There are three elimination conditions:

Elimination condition 1:

(only X1) and (only X2) and Not (X1∩ X2)

Trang 6

Elimination condition 2:

(only X1) and (every X2) and Not (X2⊆ X1) Elimination condition 3:

(only X1) and (some X2) and Not (X1∩ X2)

Consider the example The system includes a subset of three centrifugal pumps in

parallel, any two of which are continually operational, with the following specifica-tions requirement and subset interval:

Specifications:

System requirement:< all-parts only kPa 5000 10000 >

Subset interval:< all-parts every kPa 2000 10000 >

where:

Pump 1 interval:< all-parts every kPa 1000 10000 >

Computation: elimination condition 2:

(only X1) and (every X2) and Not (X2⊆ X1)

Subset interval:

System requirement: X1=< kPa 5000 10000 >

Subset interval: X2=< kPa 2000 10000 >

Elimination result:

Condition: Not (X2⊆ X1)⇒true

Description:

The elimination condition result is true in that the pressure interval of the subset

of pumps does not meet the system requirement, where

X1=< kPa 5000 10000 >

and the subset interval

X2=< kPa 2000 10000 >

A minimum pressure of the subset of pumps (kPa 2,000) cannot be less than the minimum system requirement (kPa 5,000), prompting a review of the conceptual design

Redundancy Conditions

Redundancy conditions determine if a subset’s labelled interval (X1) is not

signifi-cant because another subset’s labelled interval (X) is dominant

Trang 7

In order for the redundancy conditions to apply, the items set and the state set

of the labelled interval (X1) must be a subset of the items set and state set of the

labelled interval (X2) X1must have either an all-parts label or a some-parts label

that can be redundant with respect to X2, which in turn has an all-parts label

Redundancy conditions do not apply to X1having an all-parts label while X2has

a some-parts label Each redundancy condition is formatted so that there are two subset labelled intervals and a condition

There are five redundancy conditions:

Redundancy condition 1:

(every X1) and (every X2) and (X1⊆ X2) Redundancy condition 2:

(some X1) and (every X2) and (X1∩ X2) Redundancy condition 3:

(only X1) and (only X2) and (X2⊆ X1) Redundancy condition 4:

(some X1) and (only X2) and (X2⊆ X1) Redundancy condition 5:

(some X1) and (some X2) and (X2⊆ X1)

Consider the example The system includes a subset of three centrifugal pumps in

parallel, any two of which are continually operational, with the following specifica-tions requirement and different subset configuraspecifica-tions for the two operational units, while the third functions as a standby unit:

Specifications:

System requirement:< all-parts only kPa 1000 10000 >

Pump 1 interval: < all-parts every kPa 1000 10000 >

Subset configuration 1:

Subset1 interval:< all-parts every kPa 1000 10000 >

where:

Trang 8

where:

Computation:

(every X i )(As,i, S i)→ (every x max i xl,i mini xh,i)(A ∩ i S i)

(every X1) and (every X2) and (X1⊆ X2)

For the three subset intervals:

1) Subset intervals:

Subset1 interval: X1=< kPa 1000 10000 >

Redundancy result:

Condition: (X1⊆ X2)⇒false

Description:

The redundancy condition result is false in that the pressure interval of the pump

subset’s labelled interval (X1) is not a subset of the pump subset’s labelled

inter-val (X2)

Redundancy result:

Condition: (X1⊆ X2)⇒false

Description:

The redundancy condition result is false in that the pressure interval of the pump

subset’s labelled interval (X1) is not a subset of the pump subset’s labelled

inter-val (X2)

Redundancy result: Condition: (X1⊆ X2)⇒true

Description:

The redundancy condition result is true in that the pressure interval of the pump

subset’s labelled interval (X1) is a subset of the pump subset’s labelled

inter-val (X )

Trang 9

Subset2 and/or subset3 combinations of pump 1 with pump 3 as well as pump 2 with pump 3 respectively are redundant in that pump 3 is redundant in the con-figuration of the three centrifugal pumps in parallel

Translation Rule

The translation rule generates new labelled intervals based on various interrelation-ships among systems or subsets of systems (equipment) Some components have variables that are directional (Typically in the case of RPM, a motor produces RPM-out while a pump accepts RPM-in.) When a component such as a motor has

a labelled interval that is being considered, the translation rule determines whether

it should be translated to a connected component such as a pump if the connected components form a set with matching variables, and the labelled interval for the motor is not redundant in the labelled interval for the pump

Consider the example A system includes a subset with a motor, transmission and

pump where the motor and transmission have the following RPM ratings:

Component Min RPM Max RPM

Labelled intervals:

Motor = < all-parts every rpm 750 1500 > (normal)

Transmission= < all-parts every rpm 75 150 > (normal)

Translation rule:

Pump= < all-parts every rpm 75 150 > (normal)

Propagation Rules

Propagation rules generate new labelled intervals based on previously processed

labelled intervals and a given relationship G, which is implicit among a minimum

of three variables Each rule is formatted so that there are two antecedent subset

labelled intervals, a given relationship G, and a resultant subset labelled interval.

The resultant labelled interval contains a constraint label and a labelled interval calculus operative The resultant labelled interval is determined by applying the operative to the variables If the application of the operative on the variables can produce a labelled interval, a new labelled interval is propagated If the application

of the operative on the variables cannot produce a labelled interval, the propagation rule is not valid

An item’s set and state set of the new labelled interval are the intersection of the item’s set and state set of the two antecedent labelled intervals If both of the antecedent labelled intervals have an all-parts set label, the new labelled interval

Trang 10

will have an all-parts set label If the two antecedent labelled intervals have any other combination of set labels (such as one with a some-part set label, and the other with an all-parts set label; or both with a some-part set label), then the new labelled interval will have a some-part set label (Davis 1987)

There are five propagation rules:

Propagation rule 1:

(only X ) and (only Y ) and G ⇒ (only Range (G, X, Y ))

Propagation rule 2:

(every X ) and (every Y ) and G ⇒ (every Range (G, X, Y ))

Propagation rule 3:

(every X ) and (only Y ) and state variable (z) or parameter (x)

and G ⇒ (every domain (G, X, Y ))

Propagation rule 4:

(every X ) and (only Y ) and parameter (x) and G ⇒(only SuffPt (G, X, Y))

Propagation rule 5:

(every X ) and (only Y ) and G ⇒ (some SuffPt (G, X, Y))

Consider the example Determine whether the labelled interval of flow for

dy-namic hydraulic displacement pumps meets the system specifications requirement where the pumps run at revolutions in the interval of 75 to 150 RPM, and the pumps have a displacement capability in the interval 0.5 ×10 −3 to 6×10 −3 cubic metre

per revolution Displacement is the volume of fluid that moves through a hydraulic line per revolution of the pump impellor, and RPM is the revolution speed of the pump The flow is the rate at which fluid moves through the lines in cubic metres per minute or per hour

Specifications:

System requirement:< all-parts only flow 1.50 60 > m3/h

Given relationship:

Flow (m3/h) = (Displacement × RPM) ×C

where C is the pump constant based on specific pump characteristics.

Displacement (η)= < all-parts onlyη0.5 ×10 −36×10 −3 >

RPM (ω) = < all-parts onlyω75 150>

certain conclusions about the integrity of engineering. ..

Description of Example

In the conceptual design of a typical engineering process, most sets of systems in- clude a single process vessel that is served by a subset of three centrifugal pumps in

Định dạng
Số trang	10
Dung lượng	75,33 KB