Performance P f Estimation E ti ti Using U i Similarity Models and Design Information Entropy. Petter Krus Division of Machine Design Department of Management and Engineering Linköping University

Estimation Models for System y Design g ¾ The increasing computational resources means that large high fidelity models can be used for system analysis, so why bother with performance estimation models. ¾ High fidelity models requres a high fidelity design design, which is not available in early system level design. ¾ Estimation Models should have a predictive fidelity that is consistent i t t with ith th the d design i fid fidelity lit att that th t particular ti l stage. t ¾ Estimation models is not used directly for design of the component/subsystem they represent, merely to predict what can be achieved. This can later be used for component selection or as input for component design.

2008-08-17

Sid 2

Linköpings universitet

System y level development p p process

Identification of customer needs (compatible with capability)

Generating requirement and desirables specification

Concept generation

Concept selection

Quantitative Q tit ti refinement

Analysis A l i and d evaluation

Sensitivity and trade tradeoff analysis

2008-08-17

Sid 3

Linköpings universitet

Performance Estimation ¾ Statistical models has been used for aircraft preliminary design for a long time to estimate weight of subsystems. This is related to similarity models (or allometric models) used in biology to describe morphology depending on size. These can be combined with analytical models for scaling relations ¾ M Modells d ll can also l be b derived d i d tto correlate l t performance f with ith th the degree of refinment. Here design information entropy is a useful quantity.

2008-08-17

Sid 4

Petter Krus

Information Entropy py ¾ Design information entropy is a measure of the contraction of design space during design. It is roughly proportional to the number of objects in the design space and thus a more convenient unit than the number of possibilities excluded. ¾ Relative Information entropy is a very useful state in evolutionary l ti llearning i processes. It iis consistent i t t with ith th the view i that the design process is a learning process. ¾ A general model is presented that links information to optimization, ti i ti refinement, fi t performance f and d cost. t ¾ The results indicate that it sometimes may be more efficient to increase the number of design parameters, rather than refining a few parameters to a high degree degree.

2008-08-17

Sid 5

Petter Krus

Design Information Entropy is a Measure of the Size of the Design Space Lego example ¾ The design g space p of a set of Lego g bricks represents all combinations of arranging these bricks. ¾ With a set of only two bricks with four knobs on each there are 51 discrete possible arrangements ¾ Two of these represents picking only one brick. And one state is to p pick no one. ¾ The 51 different configuration (states) means that the amount of information needed to specify a particular design is:

I x = log 2 nDstate = log 2 51 = 5.7bits

2008-08-17

Sid 6

Petter Krus

Amount of Information content (Information Entropy) The differential information entropy for continuous signals, defined by Shannon [1] as: ∞

H = − ∫ p( x) log 2 ( p( x))dx −∞

Kullback-Leibler divergence ∞

H rel

p( x) )dx = − ∫ p ( x) log 2 ( m( x ) −∞

Generalized ∞

H rel

2008-08-17

∞

p ( x1 K xn ) = − ∫ L ∫ p ( x1 K xn ) log 2 ( )dx1 L dxn m( x1 K xn ) −∞ −∞ Petter Krus

Sid 7

Amount of Information content (Information Entropy) The distribution m(x) should be a rectangular distribution in the bounded interval.

I x = H rel ( x) = −

xmax

∫

p ( x) log 2 ( p ( x) xR )dx

xmin

Generalized x1,max

Ix = −

∫

xn ,max

L

x1,min

∫

p ( x1 K xn ) log 2 ( p( x1 ,K xn ) xR1 L xRn )dx1 L dxn

xn ,min

More compact

I x = − ∫ p (x) log 2 ( p (x) S )dS S

2008-08-17

Sid 8

Petter Krus

Amount of Information content (Information Entropy) The information content I of a variable (in bits).

xmax

I = − ∫ p( x)log2 ( p( x) xR )dx

xR = xmax − xmin

xmin

If the range xr is divided in equal parts Δx the amount of information is:

xr 1 I = log2 = log2 Δx δx

or more general:

Δx

I = log2

Here Δx is the tolerance in x.

xmin 2008-08-17

Sid 9

Petter Krus

x1

x2

xmax

S s

Design g Space p with Both Continuous and Discrete Variables ¾ The position of the inserted axis represents a continuous variables py ¾ The information entropy associated with that is dependent on the accuracy with which it is specified.

I x = log l 2 nDstates + log l nCstates + log l 2 ¾ The axis can be in three position and If the position of th axis the i within ithi one h hole l iis specified within 10% the total information entropy is:

xR Δx

1 I x = log l 2 51 + log l 2 3 + log l 2 = 10.6bits bi 0.1

2008-08-17

Sid 10

Petter Krus

x

Design Information entropy ¾ The design g information entropy y represents a measure of the precision by which a design is defined relative to the design space in consideration. ¾ It is also proportional to the dimensionality of the design problem. ¾ It can also be seen as a measure of a probability that a random design in the design space is found, that is within the given t l tolerance region. i ¾ Design information entropy, as defined here, should not be taken as a direct measure of complexity. A very simple design can represent a large information content if it has been picked from a large design space.

2008-08-17

Sid 11

Petter Krus

Design g space p expansion p ¾ The design information entropy can be increased in two ways ¾ Refinement ¾ Design D i space expansion i

¾ Design space can be increased in several ways like: ¾ Adding more bricks ¾ Adding other types of bricks ¾ Releasing more design parameters in a design

xR′ xR xR′ xR xRn xR I x = log 2 = log 2 = log 2 = n log 2 Δx Δx xR Δx xR Δx = nI p

2008-08-17

Sid 12

Petter Krus

Growth of design g information entropy py during the design process I R = log 2

Design space expansion

S0′ S0

S0′

m

∑ si

n

∑s

i =1

Design space generation

S0

i =1

Concept generation m

I I = − log 2

∑ si i =1

S0

Concept screening

Sid 13

Petter Krus

Concept optimization and selection

n

I II = − log

∑s

i =1 2 m

j (i )

∑s i =1

2008-08-17

j (i )

i

I III = − log2

sδ n

∑s i =1

j (i )

sδ

Influence factor of design parameters

System characteristics Generalised system performance

y = f x ( x)

z = g(y) z = z( x)

ψ (i )= z i ( xR ,i , x p ,opt ,i ) − z i −1 ( xR ,i −1 , x p ,opt ,i −1 )

An aproximate modell of this is:

ψ (i)= c pi 2008-08-17

Sid 14

Petter Krus

−λ

Total influence up to the n:th parameter: n

n

i =1

i =1

Ψ ( n)= ∑ψ (i ) = c p ∑ i

−λ

= c p H ( n, λ )

H ( n, λ )

(

is the harmonic function

)

zopt ( n ) = z n x p ,opt = Ψ (n) − z0 = n

= cp ∑ i i =1

−k

−z0 = c p H (n, λ ) − z0

⎛ 1 ⎛ 1 ⎞⎞ ≈ c p ⎜ c (λ ) + ⎜1 − λ −1 ⎟ ⎟ − z0 λ −1 ⎝ n ⎠ ⎠ ⎝ 2008-08-17

Sid 15

Petter Krus

Asymptotic y p behaviour

(

)

zopt ( n ) = z n x p ,opt = Ψ (n) − z0 = n

= c p ∑ i − k −z0 = c p H (n, λ ) − z0 i 1 i=

⎧ ∞ when λ ≤ 1 lim H (n, λ ) = ς (λ ) = ⎨ n →∞ ⎩finite value > 1 ς (λ )

4 3.5 3 2.5 2 1.5 1 0.5 2

2008-08-17

Sid 16

Petter Krus

3

4

λ

5

Sorted spectrum of parameter influence

ψ (i)= c p i

−λ

λ =1

Influence

12 10 8 6 4 2 0 1

2

3

4

5

Figure 3. Sorted parameter influences

2008-08-17

Sid 17

Petter Krus

6

7

8

9

Total influence up to the n:th parameter: n

n

i =1

i =1

Ψ ( n)= ∑ψ (i ) = c p ∑ i

−λ

= c p H ( n, λ )

H ( n, λ )

(

is the harmonic function

)

zopt ( n ) = z n x p ,opt = Ψ (n) − z0 = n

= cp ∑ i i =1

−k

−z0 = c p H (n, λ ) − z0

⎛ 1 ⎛ 1 ⎞⎞ ≈ c p ⎜ c (λ ) + ⎜1 − λ −1 ⎟ ⎟ − z0 λ −1 ⎝ n ⎠ ⎠ ⎝ 2008-08-17

Sid 18

Petter Krus

Accumulated influence ⎧ ∞ when λ ≤ 1 lim H (n, λ ) = ς (λ ) = ⎨ n →∞ ⎩finite value > 1 40

18

35

16

30

14 12

25

10 20

λ =1

15 10 5 0

8

4 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

80 70 60 50 40

λ = 0.5

30 20 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2008-08-17

λ=2

6

Sid 19

Petter Krus

Performance model

z ( n ) = k p H ( n, λ ) − z 0 zopt (n) − zopt ( n, δ x ) = k p H ( n, k )δ x

δx = 2

−

Performance loss due to uncertaintyy

Ix n

Ix n

zopt (n, I x ) = k p H (n, k )( )(1 − 2 ) − z0

2008-08-17

Sid 20

Petter Krus

Example: optimization of a beam

F 1

2

...........

n

L

FL m= η kσ max 2008-08-17

Sid 21

Petter Krus

η ∈ ]0,1]

Efficiency y as a function of information 0.9 η

5 segments

0.8 0.7 0.6 1 segment 0.5

5

10

15

20

25

Ix

30

Figure 6. The efficiency of the structure as a function of information Ix

2008-08-17

Sid 22

Petter Krus

The optimal number of design parameters as a function of design information entropy

n

λ=1

20

15

Bounded performance

Unbounded performance

10

λ=2 5

20

2008-08-17

Sid 23

Petter Krus

40

60

80

Ix

100

Influence values

ψ (i )= c p i − λ 07 0.7 0.6 0.5 0.4

Influence

0.3

Model

02 0.2 0.1 0 1

2

3

4

5

6

7

8

9

Figure 7. Influence of increasing the number of elements in the beam model. Exact solution and a model based on a negative power law relationship with λ=2.

2008-08-17

Sid 24

Petter Krus

Influence values

ψ (i)= c pi −λ

9.00 8.00 7.00 6.00 5 00 5.00

I fl Influence values l

4.00

Adjusted model

3.00 2.00 1.00

tc

t

ise Vc ru

C

T

r C

W e

W f

xw

Bh t

B

0.00

Figure 10. Sorted influences for different design parameters of a transport aircraft aircraft. Adjusted model with λ=1.4. =1 4

2008-08-17

Sid 25

Petter Krus

Performance model Ix n

zopt (n, I x ) = c p H (n, λ )(1 − 2 ) − z0 I x = C / kc

Substituting information with cost

zopt (n, C ) = c p H (n, λ )(1 − 2

C / kc n

) − z0

Or as a continuous function

zopt (n, I x ) = c′p 2008-08-17

Sid 26

Petter Krus

1 n

(1 − 2 λ −1

C kCI n

) − z0′

4

4

3.5

3.5

3

3

performance [G Hz eqv]

performance [GHz eqv]

Example p

2.5 2 1.5 1 0.5

2.5 2 1.5 1 0.5

0

0

0

1000

2000

3000

price [SEK]

4000

5000

0

1000

2000

3000

4000

price [SEK]

Processor performance as a function of price for Intel Pentium and Celeron processors (2004), with fitted model

2008-08-17

Sid 27

Petter Krus

5000

Modelling of components/subsystem for system level design Design parameters

Reduced design parameters

ii.e. e max torque torque, speed, etc

Secondary characteristics

i.e. weight, volume, cost, power loss

Component model

i.e. geometrical dimensions

Functional characteristics h t i ti

Functional (primary) characteristics

Allometric All ti Component model

Allometric models are derived under Secondary the assumption that the design characteristics parameters t in i the th component/system are optimized with respect to the secondary characteristics (Design parameters)

General Power Transformation i.e electric motors, hydraulic motors, mechanical gears, etc

e1 f1

Power transformer

e2 f2

Power losses e

are effort variables i.e. voltage, force, pressure

f

are the flow variables ii.e. e current current, speed speed, flow

P=ef

Modelling of components/subsystem for system level design

Secondary characteristics

Functional characteristics

Mass, m

Max power, P

Volume V Volume, Max torque, T

Estimation model

Mom. inertia, J Efficiency Cost Life Reliability

Table of hydraulic y motor data

displacem max ent power [cm3] 59 00 59.00 103.00 5.00 10.00 14.00 19.00 22.00 28.00 40.00 56.00 71.00 125.00 250.00

max torque

[W] [Nm] 44000 331 00 331.00 62000 572.00 24730 33.00 45338 66.00 60142 96.00 71919 127.00 62000 140.00 79000 178.00 106000 255.00 134000 357.00 132000 509.00 190000 895.00 321000 1790.00

volume [cm3] 2203 3076 742 1082 1282 1683 1596 1596 2738 2374 5967 9389 20838

diameter [mm] 117 123 84 94 102 114 118 118 150 133 170 200 265

mass

moment of inertia

[kg]

[kg cm2]

85 8.5 12.50 5.00 7.5 8.3 11.00 11.00 11.00 15.00 21.00 34.00 61.00 120.00

1.6 3.9 4.2 11.00 15.00 15.00 43.00 85.00 121.00 300.00 959.00 Average values

power intensity kp [kW/kg]

5 18 5.18 4.96 4.95 6.05 7.25 6.54 5.64 7.18 7.07 6.38 3.88 3.11 2.68 5.45

mean pressure (torque power torque density) density intensity pm kT ρp [bar] [W/cm3] [Nm/kg] 1 50 1.50 19 97 19.97 38 94 38.94 1.86 20.16 45.76 0.44 33.33 6.60 0.61 41.90 8.80 0.75 46.91 11.57 0.75 42.73 11.55 0.88 38.85 12.73 1.12 49.50 16.18 0.93 38.71 17.00 1.50 56.44 17.00 0.85 22.12 14.97 0.95 20.24 14.67 14.92 0.86 15.40 1.00

34.33

17.74

Inertia fraction

εJ

NA NA 0.018 0.024 0.019 0.031 0.039 0.039 0.051 0.092 0.049 0.049 0.046 0.042

Table of electric motor data

Voltage [V] 8.4 8 24 24 24 50 460 460 460

max power [W] 210 320 609 1440 3580 15992 73763 198499 491751

speed at load

max torque

[rad/s] [Nm] 1068 0.20 1378 0 23 0.23 523 1.164 450 3.2 471 7.6 419 38.20 175 420.74 215 922.87 175 2813.33

volume

mass

[cm3]

[kg] 0.1716 0 29 0.29 1.10 2.4 3.9 9.36 215.00 215.00 907.00 Average values

55 54 343 729 729 4539 23487 86524 165518

power intensity kp [kW/kg] 1.22 1 10 1.10 0.55 0.60 0.92 1.71 0.34 0.92 0.54 0.88

mean pressure (torque power torque density) density intensity pm kT ρp [bar] [W/cm3] [Nm/kg] 0.04 3.84 1.15 0 04 0.04 5 95 5.95 0 80 0.80 0.03 1.78 1.06 0.04 1.98 1.33 0.10 4.91 1.95 0.08 3.52 4.08 0.18 3.14 1.96 0.11 2.29 4.29 3.10 0.17 2.97 0.09

3.38

2.19

Table of planetary yg gearbox data

max power

max torque

[W] [Nm] 329 92 550 176 1125 372 1272 304 1333 38 2543 607 6283 110 6597 1000 18473 560 18850 1440 18850 1440 28589 1820

gear ratio

51 51 51 50 9 50 10 50 10 10 10 20

volume [cm3] 152 729 1331 975 452 1610 462 2839 2162 4180 4180 2185

diameter [cm] 5.7658 9 11 3.54 8.001 11.5062 7 13 13 16 16 14

mass

moment of inertia

power intensity kp

[kg]

[kg cm2]

[kW/kg] 0.66 0.79 0.70 0.61 1.11 0.94 2.17 0.44 1.61 0.70 0.70 1.86

0.5 0.7 1.6 2.1 1.2 2.7 2.9 15 11.5 27 27 15.4

0.38 4.96 4.1 12.3 12.3 6.85 Average values

1.02

mean pressure (torque density) pm

power density

ρp

[bar] [W/cm3] 6.05 2.16 2.41 0.75 2.79 0.85 3.12 1.30 0.84 2.95 3.77 1.58 2.38 13.60 3.52 2.32 2.59 8.54 3.44 4.51 3.44 4.51 8.33 13.08 3.56

4.68

torque intensity kT

Inertia fraction

εJ

[Nm/kg] 184.00 251.43 232.50 144.76 31.67 224.81 37.93 66.67 48.70 53.33 53.33 118.18

NA NA NA NA NA NA 0.0107 0.0078 0.0084 0.0071 0.0071 0.0091

120.61

0.008

Moment of inertia All mass is contributing g to the moment of inertia and located at the diameter of the machine

⎛d ⎞ J max = m⎜ ⎟ ⎝2⎠

d

2

The inertia fraction is defined as the ratio between the actual moment of inertia, and the theoretical max value.

εJ =

J J max

=

J ⎛d ⎞ m⎜ ⎟ ⎝2⎠

2

hence

2 ) d )⎛ ⎞ J = ε J m⎜ ⎟ ⎝ 2⎠

Estimation of motor characteristics Mass, m Max power, P Max torque, T

Diameter, d

) m = P / ρP ) V = T / pm 2 ) d )⎛ ⎞ J = εJm⎜ ⎟ ⎝2⎠

Volume, V

Mom. inertia, J

ρP, pm, and eJ are technology parameters. They are reasonably constant for a technology regardless of size size.

Conservative mass and volume estimation) ) ) m = max((mP , mT ) ) ) ) V = max(VP , VT ) 2 ) )⎛ d ⎞ J = ε J m⎜ ⎟ ⎝2⎠

where

) mP = P / ρ P ) VP = P / k P ) mT = T / kT ) VT = 2πT / p m

Multiple p regression g models ) log m = a10 + a11 log P + a12 log T ) log V = a 20 + a 21 log P + a 22 log T ) l J = a30 + a31 log log l P + a32 log l T This can also be written as

) a10 a21 a22 Vs = 10 P T ) m s = 10 a10 P a11 T a12 ) a30 a31 a32 J s = 10 P T

Note: N t Becomes B zero when P or T is zero (unrealistic)

Estimation model of mass for hydraulic motor Mass relation

Mass relation 3

2 Y Predicted Y 1

log(mass [kg])

log g(mass [kg]

3

2 Y Predicted Y

1

0

0 4

5 log(power [W])

6

1

2

3

log(torque [Nm])

4

Mass and volume estimation Hydraulic H d li motor

) ms = 7.26 ⋅10 −5 P 0.99T 0.202 ) Vs = 0.53 ⋅10 −6 P 0.493T 0.53 ) −0.734 2.12 J s = 1.34 P T This can be used together with the other expressions for a conservative estimation

) ) ) ) V = max(Vs ,VP ,VT ) ) ) ) ) m = max(ms , m P , mT ) 2 ) ) )⎛ d ⎞ J = max( J s , ε J m⎜ ⎟ ) ⎝2⎠

Alternative form ⎛P⎞ ) m = m0 ⎜⎜ ⎟⎟ ⎝ P0 ⎠

a11

) ⎛P⎞ V = V0 ⎜⎜ ⎟⎟ ⎝ P0 ⎠

a 21

) ⎛P⎞ J = J 0 ⎜⎜ ⎟⎟ ⎝ P0 ⎠

a31

⎛T ⎞ ⎜⎜ ⎟⎟ ⎝ T0 ⎠

a12

⎛T ⎞ ⎜⎜ ⎟⎟ ⎝ T0 ⎠

a 22

⎛T ⎞ ⎜⎜ ⎟⎟ ⎝ T0 ⎠

Can be used for scaling of an existing motor

a32

Model of mass as a function of power ) m = 7.26 ⋅10 P (h d li motor) (hydraulic t ) −5

s

0 99 0.99

0 202 T 0.202

) Vs = 0.53 ⋅10 −6 P 0.493T 0.53

) J s = 1.34 P −0.734T 2.12 Mass[kg]40

1000

20

800

0

600 400

50000 100000

200 150000

Power[W]

200000 0

Torque [Nm]

) ) ) ) m = max(ms , mP , mT )

Model of volume as a function of power ) m = 7.26 ⋅10 P T (h d li motor) (hydraulic t ) −5

0 99 0.99

0.202 0 202

s

) Vs = 0.53 ⋅10 −6 P 0.493T 0.53

) J s = 1.34 P −0.734T 2.12 8000 6000 4000 2000 0

Volume [cm3]

1000 800 600 Torque [Nm]

400

50000 100000 Power[W]

200 150000 200000 0

) ) ) ) V = max(Vs , VP , VT )

Main axis of data points (electric motor) "Line of natural ratio" y = 1.2696x - 3.6972 R2 = 0.9881

3 2

Series1 Linear (Series1)

1 0 1

2

3

4

5

6

-1 log(Power[W]) 4 lo og(torque[Nm])

log g(Torque[Nm])

4

Hydraulic

y=1 1.3805x 3805x - 4.4009 4 4009 2 R = 0.7225

3 2

Series1 Linear (Series1)

1 0 1

2

3

4

-1 log(powe r[W])

5

6

Principle p Component p Analysis y ¾ Introduces new set of artificial design parameters. Could be close to real parameters such as ”size” and aspect ratio ¾ Removes the need to handle the extreems. ¾ Useful for minimizing waste in the design space. ¾ It also differnentiate the impact p of the design g p parameters to a maximum degree. (Maximizes the decay exponent λ)

log(T ) log( x2 )

log( x2 )

log( x1 )

4

4.00

3

3.00

2

2.00

1

1.00

0 0

1

2

3

4

6

7

0.00 0.00

-1

-1.00

-2

-2.00

-3

-3.00

log( P ) 2008-08-17

5

Sid 44

Linköpings universitet

1.00

2.00

3.00

4.00

log( x1 )

5.00

6.00

7.00

Estimation models using PCA design parameters (2-parameter case) Size

g( P) ⎞ ⎛ cos ϕ ⎛ log( ⎜ ⎟=⎜ ⎝ log(T ) ⎠ ⎝ sin ϕ

, x1,max ] g( x1 ) ⎞ x1 ∈ [ x1,min − sin ϕ ⎞ ⎛ log( , , ⎟ ⎟⎜ cos ϕ ⎠⎝ log( x2 ) ⎠ x2 ∈ [ x2,min , x2,max ]

) a10 Vs = 10 P a21 T a22 ) m s = 10 a10 P a11 T a12 ) a30 a31 a32 J s = 10 P T

2008-08-17

Sid 45

Linköpings universitet

Auxiliary parameter 1

Modelling for system level design Performance parameters (PCA-set) i.e. size

Allometric component model

Functional (primary) characteristics

ii.e. max ttorque, speed, etc

Secondary characteristics

i.e. weight, volume, cost, power loss

Allometric (similarity) models are derived under the assumption that the design parameters in the component/system are optimized with respect to th secondary the d characteristics. h t i ti Models based on PCA-parameter sets are useful for trade-of studies and system optimization

Using g component data directly y

Voltage [V] 8,4 8 24 24 24 50 460 460 460

max power [W] 210 320 609 1440 3580 15992 73763 198499 491751

speed at load

max torque

[rad/s] [Nm] 1068 0,20 1378 0,23 523 1,164 450 3,2 471 7,6 419 38,20 175 420 74 420,74 215 922,87 175 2813,33

volume [cm3] 55 54 343 729 729 4539 23487 86524 165518

mass [kg] 0,1716 0,29 1,10 2,4 3,9 9,36 215 00 215,00 215,00 907,00

power intensity kp [kW/kg] 1,22 1,10 0,55 0,60 0,92 1,71 0 34 0,34 0,92 0,54

mean pressure (torque power torque density) density intensity pm kT ρp [bar] [W/cm3] [Nm/kg] 0,04 3,84 1,15 0,04 5,95 0,80 0,03 1,78 1,06 0,04 1,98 1,33 0,10 4,91 1,95 0,08 3,52 4,08 0 18 0,18 3 14 3,14 1 96 1,96 0,11 2,29 4,29 3,10 0,17 2,97

¾ Component data is increasingly being made available from the internet through Globalspec, eCl@ss etc, and to a greater extent machine readable. y level design g ¾ This makes use of real data for system optimization possible.

Design optimization using component data directly Design on demand, Micro Aerial Vehicles, MAV

Baseline

Optimized

motor

Astroflight 010

Hacker B20 31S

propeller

Graupner 6x3 folding APC 5x5

motor controller

Astroflight 10

YGE 4

Battery

Etec 1200 2s1p

Tanic 1100 3s1p

S AR

MAV baseline

P

P

b

d

ηd

ηb

MAV propulsion l i system.

0.12 m2

0.12

1.33

1.33

Endurance at cruise

22 min

Max speed T/W ratio

75 km/h 0 95 0.95

Pm ηm

42 min 12 1.2

Pout ηp

ηtot=ηb*ηd*ηm *ηp

MAV optimized geometry

Result: Shopping list and geometrical dimensions

David Lundström, Petter Krus

Conclusions ¾ Models for performance prediction can be based on statistics from existing subsystems and components is a simple and well established practice although it has had limited use outside aircraft design design, and is not present in the university curriculum to the extent it deserves. ¾ Design information entropy can be used to describe the impact of further refinement and design space expansion. This is a promising area for further study. Also potential for forecasting of performance ¾ Principal Component Analysis PCA can be used to produce parameters sets for models suitable for system optimization. It produces design spaces that minimize waste of the design space space.

2008-08-17

Sid 49

Linköpings universitet

References

2008-08-17

¾

P. Krus. “Estimation Models for Concept Optimisation of Fluid Power Transformation and Transmission”. The Ninth Scandinavian International Conference on Fluid power, SICFP’05, June 1-3, 2005, Linköping, Sweden. http://petter.krus.googlepages.com/sicfp05Krus4.pdf

¾

Torenbeek E, (1982), Synthesis of subsonic design, Delft University Press Kluwer Academic Publisher, ISBN 90-247-2724-3. 1982.

¾

Raymer y D.P, ((1992), ) Aircraft Design: g A conceptual p approach, pp AIAA Education series, ISBN 0-930403-51-7. 1992.

¾

Calder W.A, (1996), Size function, and life history, Dover publications, inc. Mineola, New York. 1996.

¾

P Krus, A Jansson, J-O Palmberg, (1991), Optimization for Component Selection in Hydraulic Systems, 'Fourth Bath International Fluid Power Workshop', Bath, UK 1991

¾

Shannon, D., “A mathematical theory of communication”, Bell Syst. Tech. J. 27 379 , 1948.

¾

Ullman, D. G. “The Mechanical Design Process”. McGraw-Hill Book Co, Singapore 1992. ISBN 0-07-065739-4.

¾

Kullback, S., and Leibler, R. A., “On information and sufficiency”, Annals of Mathematical Statistics 22: 79-86. 1951.

¾

Chaitin, G, “Algorithmic Information Theory”, Cambridge University Press, 1987.

¾

Ming Li, Paul Vitányi. “An Introduction to Kolmogorov Complexity and its applications”. Springer Verlag 1997. ISBN 0-387-94868-6.

¾

Frey, D. Jahangir, E., “Differential entropy as a measure of information content in axiomatic design”. Proceedings of the 1999 ASME Design engineering technical conference, Las Vegas, USA, 1999.

¾

Box M Box, M. JJ.. “A A new method of constrained optimization and a comparison with other methods” methods . Computer Journal Journal, 8:42--52 8:42--52, 1965 1965.

¾

Krus P, A Jansson, J-O Palmberg, “Optimization Using Simulation for Aircraft Hydraulic System Design”, Proceedings of IMECH International Conference on Aircraft Hydraulics and Systems, London, UK, 1993

¾

Khan, W. A. and Angeles, J., “The role of entropy in design theory and methodology”, Proc. CDEN/C2E2 2007 Conference, Winnipeg, Alberta, Canada 2007

¾

Utterback J M, “Mastering the Dynamics of Innovation”. Harvard Business School Press, Boston, USA, 1994.

¾

Bennett, C, “Dissipation, Information, Computational Complexity and the Definition of Organization”. Emerging Synthesis in Science, (ed.) D. Pines, (Boston: Addison-Wesley, 1985).

¾

Suh, N.P., A “Theory of Complexity, Periodicity and the Design Axioms”. Research in Engineering Design 1999. Springer-Verlag.

¾

Bras, B. and Misstree, F., “Compromise Design Decision Support Problem for Axiomatic and Robust design”, Journal of Mechanical design, Transactions of the ASME. V117, 1995.

¾

Krus P, J Andersson, “An information theoretical perspective on design optimization”, ASME, DETC, Salt Lake City, USA, 2004.

¾

Hatchuel, A. and Weil, B., “A new approach of innovative design: an introduction to C-K theory”, Proceedings International Conference on Engineering Design – ICED 03, Stockholm, Sweden, 2003.

¾

Hick W. E. “On the rate of gain of information". Quarterly Journal of Experimental Psychology, 4:11-26, 1952.

Sid 50

Linköpings universitet