July 28, 2020

Silent Engineering(Lecture 6)

Structural Optimization to ReduceSound Radiation Power- Noise reduction by adding ribs or

hollows on the plate -

Tokyo Institute of TechnologyDept. of Mechanical EngineeringSchool of Engineering

Prof. Nobuyuki Iwatsuki

１. Various Optimization Methods

Optimization method:Numerical calculation to obtain design variables to make an objectivefunction minimize or maximize

.max.min)( 　　　>−XΦ orObjective function

Design variable

Constraints:

UL XXXXX

=

0)(0)(

ΨΨ

Inequality constraints

Equation constraints

Constraints on design variables

1.1 Design variables, objective function and constraints

X1

X2

Φ(X1,X2)

1.2 Principles of various optimization methodsClassification of methods:(1) Linear optimization

・Linear programing (LP)Suitable for problems with objective function and constraintswhich are described as linear equations with respect to designvariables

(2) Nonlinear optimization1)Conventional methods○Gradient method

To Search minimum/maximum value in a direction calculatedas gradient of the objective function

Suitable for Logistics planning, Production planning

・The steepest descent method“The most traditional method”

・Davidon-Fletcher-Powell method“Various improvements”

・Newton’s method“Partial derivative equation of objective function”

2)New methods・Genetic algorithm （GA)

“A method adapting natural selection of creature”・Simulated annealing (SA)

“A method adapting behavior of metal molecular inannealing process”

・Particle Swarm Optimization (PSO)“A method adapting behavior of swarm of bird”

○Direct search methodTo directly search minimum function value

・Downhill simplex method“To search optimum value with a simplex composed ofa several combinations of design variables”

(1)The steepest descent methodThe descent direction vector of the objective function at a current Point, X, can be calculated as

)(grad)(

21

XX

Xd ΦΦΦΦΦ −=

∂∂

−∂∂

−∂∂

−=∂

∂−=

T

nXXX (1)

The minimal point in the descent direction is then searched with an one-dimensional search.

X2

X1

|d|

Φ

Xi

Xi+1di

Search with the steepest descent method

The recurrence formula ofthe design variable vector:

iii XdX +=+ α1 (2)

One-dimensional search:

For example, the quadratic function passing through 3 points which are currentvariable, di, and other 2 points, da and db, should be calculated and its minimal pointis assumed as the next point.

Xi

Φ

|d|

Xi+1 XbXa

αi αbαi+1 αa

Actual functionApproximatedquadratic function

2210

2210

2210

bbb

aaa

iii

cccΦcccΦ

cccΦ

αα

αα

αα

++=

++=

++=

=

b

a

i

bb

aa

ii

ΦΦΦ

ccc

-1

2

2

2

2

1

0

111

αααααα

(3)

iiii

ii cccΦ

cc

ccc

ccccccdΦ

dXX 1112

21

012

11

2

21

0

2

2

12

2210

4,

2

42)(

+++

++

+=

−=−=∴

−+

+=++=

α

α

ααα

　(4)

Strong points:○Convergence is very quick.○High accuracy

Primary difference can be used as approximate derivative.

Weak points:▲Derivative of function required.

(2)Downhill simplex method

A simplex which vertices are design variables with a small distances will be refreshed and directly search minimum value of the objectivefunction.

X1

X2

(X1,i, X2,i) (X1,i+ε1,i , X2,i)

(X1,i , X2,i+ε2,i)

(X1,i+1, X2,i+1)

Simplex=Polyhedron with n+1 vertices

Simplex

Refresh of simplex:

Xmax

Xmax

Xmax

XG

Xi+1

Xi+1

Xi+1

XG

XG

α

1+α

1−β β

γ1−γ XR

Reflection

Extension

Contraction

maxGR1 )1( XXXX α++==+ ai

Gmax1 )1( XXX ββ −+=+i

GR1 )1( XXX γγ −+=+i

(5)

(6)

(7)

where

maxG

max

X XX : Vertex which takes maximum function

: COG of vertices excluding

X1

X2

Iteration process of the downhill simplex method

Initial simplexExtension

ContractionReflection

Initial

Optimum

Weak points:▲Convergence is slow.

Strong points:○Not divergent○Derivative is not required.○Simple principle

(3)Genetic Algorithm (GA)Many genes each of which is composed of serially connected binarydata of design variables are evaluated on adaptability and are selected.The generation of genes should then be refreshed and finally the optimum design variables can be obtained.

X1 …. Xn

1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 1… …

×M genes in a generation

Evolution process:・Crossover・Mutation

Selected based on adaptabilitywhich is defined with objective function

Evolution process

Strong points:○Global minimum can be obtained.○Derivative is not required.

Weak points:▲Because of length of binary data of design variables, the accuracy

is not so high.▲Convergence is slow.

1.3 Example of calculation

Tested function:2

12

22

121 )1()(2 XXXXX −+−=),(Φ

The exact solution:

)0.1,0.1(21 =),( XX

The initial design variable (The steepest descent method and downhill simplex method):

)0.3,0.8(21 =),( XX

The iteration criterion(The steepest descent method and downhill simplex method):

)10,10( 1212functionVariable−−=),( εε

(8)

Genetic algorithm:Population : 200 genes Generations : 4000

Φ=4

000

Φ=4

000

Φ=1

000 Φ=1

000

Φ=100

Φ=100

Φ=10

Φ=1

Initial

Final

Optimization result obtained with the steepest descent method

Φ=4

000

Φ=4

000

Φ=1

000 Φ=1

000

Φ=100

Φ=100

Φ=10

Φ=1

Initial

Final

Optimization result obtained with the downhill simplex method

Φ=4

000

Φ=4

000

Φ=1

000

Φ=1

000

Φ=100

Φ=100

Φ=10

Φ=1

Genes

Initial individuals in the genetic algorithm

Genes

Optimization results obtained with the genetic algorithm

Φ=4

000

Φ=4

000

Φ=1

000 Φ=1

000

Φ=100

Φ=100

Φ=10

Φ=1

Minimum

4000thgeneration

Results:

The steepest descent

Downhill simplex Genetic algorithm

Solution (0.99999,0.99999) (1.00000,1.00000) (0.99608,0.99243)Minimum function 4.747X10-12 0.000 1.555X10-4

Iteration 77 134 4000 generationNumber of

Function calls616 268 800000

Performance of the optimization method

・The steepest descent method needs function calls to calculateprimary difference.

・The downhill simplex method is fairly effective.・Genetic algorithm needs computational time.

2. Structural Optimization to Reduce Sound Radiation Power by Adding Ribs

A cantilever plate

2.1 Design object– Rectangular cantilever plate subjected to point excitation

a

b

Thickness : h

Excitation: P

(x0,y0)

x

y

O

2.2 Estimation of sound power radiating from simple cantileverplate with the Rayleigh-Ritz method

xa

b

O

Thickness: hDensity: ρYoung’s modulus: EPoison’s ratio: ν

Boundary conditions:

0,0;,0,

0,0;0

====

=∂∂

==

QMbyaxxwwx

　

　

(9)Resultantly we obtain

0,0;,0

0,0;

0,0;0

3

3

2

2

3

3

2

2

=∂∂

=∂∂

=

=∂∂

=∂∂

=

=∂∂

==

yW

yWby

xW

xWax

xWWx

　

　

　

(10)

y

−−−⋅

−−−=

⋅=

∑

∑

by

by

by

by

ax

ax

ax

ax

C

yWxWCyxW

jyjyjy

jyjy

ixixix

ixix

jiij

jyji

ixij

,,,

,,

,,,

,,

,

,,

,

sinsinhcoscosh

sinsinhcoscosh

)()(),(

λλα

λλ

λλα

λλ

　　　　　　　　

　　　　

Eigenfunction:

i λx,i

1 1.875

2 4.694

3 7.855

4 10.996

5 :

i λy,i1 -

(Rigid motion)2 -

(Rigid motion)3 0.473

4 7.853

5 :

x-direction: Eigenfunction of

cantilever beam

(11)

y-direction: Eigenfunction ofboth ends free beam

Calculated natural frequencies and mode shapes (1)

1st mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

Calculated natural frequencies and mode shapes (2)

7th mode

8th mode

9th mode

10th mode

11th mode

12th mode

One example of the estimated sound radiation power

0.150.100.050.0010-1

10-3

10-5

10-780

60

40

20

70

50

30

10

10-2

10-4

10-610-7

FOR

CE

P RM

SN

DR

IVIN

G P

OIN

TM

OBI

LITY

Re(

Y)m

/Ns2

INPU

T PO

WER

Win

dB

ref 1

pW

SOU

ND

PO

WER

Wra

ddB

ref 1

pWLO

SS F

ACTO

RS

η dis,

ηra

d

2.3 Estimation of sound power radiating from cantilever platereinforced with straight ribs

Thickness : h

Excitation: P

(x0,y0)

x

y

O

A straight ribwith a rectangularcross-section

Rayleigh-Ritz method to calculate natural mode of vibrationof plate with straight ribs:

−−−⋅

−−−=

⋅=

∑

∑

by

by

by

by

ax

ax

ax

ax

C

yWxWCyxW

jyjyjy

jyjy

ixixix

ixix

jiij

jyji

ixij

,,,

,,

,,,

,,

,

,,

,

sinsinhcoscosh

sinsinhcoscosh

)()(),(

λλα

λλ

λλα

λλ

　　　　　　　　

　　　　

Same eigenfunction for simple plate can be assumed as

The kinetic energy and potential energy of added Euler beamsaccording to deformation of simple plate can be added to those of simple plate then Rayleigh-Ritz method can be executed.

(12)

Validation of modal analysis with experimental modal analysis

500 300

5

250

2525

(a) Simple plate

(b) y-dir. ribbed plate

(c)x-dir. ribbed plate

Mode No.

Calculated[Hz]

Measured [Hz]

1 17.14 17

2 62.64 63

3 106.71 104

4 208.91 208

5 295.06 289

6 333.37 341

7 417.50 412

8 483.86 485

9 594.14 575

10 709.31 699

11 742.05 731

Mode No.

Calculated[Hz]

Measured [Hz]

1 17.08 --

2 61.08 55

3 106.62 100

4 203.54 189

5 308.27 281

6 319.77 347

7 414.20 389

8 505.42 492

9 581.55 566

10 681.75 671

11 779.42 725

Mode No.

Calculated[Hz]

Measured [Hz]

1 19.03 --

2 63.48 55

3 119.87 112

4 216.70 198

5 326.64 296

6 338.44 347

7 448.22 420

8 484.57 471

9 661.55 643

10 740.45 721

11 780.43 737

(a) Simple plate (b) y-dir. ribbed plate (c)x-dir. ribbed plate

The calculated natural frequencies almost agreewith the measured values.

2.4 Structural optimization of cantilever plate with #-figured ribs

Thickness : h

Excitation: P

xO

y

bX2/2

a(1-X1)/2

Symmetrical #-figures rib arrangement

Design object and design variables

Objective function

Summation of peak sound powers at lower 5 natural frequencies

∑=

=Φ5

1,21 ),(

iiradWXX (13)

Optimization method

Downhill simplex method

Iteration process

Initial value

Optimum value

Variation of natural frequencies

Iteration of objective function

Total

Wrad,1

Wrad,2

Wrad,3

Wrad,4

Wrad,5

Initial value Optimum value

2.5 Structural optimization of peripherally clamped rectangular plate by adding hollows

Finite Element Model

Automatic meshing of rectangular plate with a rectangular hollow

Height of hollow is 3mm

DrivingpointBecause it is difficult

to analyze vibrationwith the R-R method,we thus use FEM.

Examples of calculation of sound radiation power

４．４ｄB

It was confirmed that a hollow reduce 4.4dB without weight increment.

Flat plate

Initial arrangement

0 100 200 300 400 500 60040

60

80

100

120

11dB reduced

Optimum arrangement

Optimization of hollow arrangement by using the gradient method with FEM vibration analysis in order to reduce noise at the 2nd mode

It is effective to arrange a nodal line on driving point at 2nd mode

3. Concluding remarksStructural optimization to reduce sound power radiating from thin plate is introduced.(1)Several optimization methods are introduced

and principles of the steepest descent method, downhill simplex method and genetic algorithmare explained.

(2)Vibration of plate adding ribs can be analyzedby adding kinetic and strain energies in theRayleigh-Ritz method.

(3)Examples of structural optimization of thin rectangular cantilever plate by adding #-figured ribs and peripherally clamped rectangular plate by adding a hollow are explained.

Subject of final report Calculate the frequency spectrum of sound power radiating froma thin/thick plate or shell under the following conditions.(1)You can arbitrarily choose vibrating plate or shell.(2)You are expected to execute modal analysis, forced vibration

analysis and estimation of sound radiation power howeveryou can assume any mode shapes or natural frequencies if it will be difficult to calculate exact or approximated vibrationmodes.

(3)You can select mechanical point excitation or sound excitationwith arbitrary frequency spectrum of excitation force.

(4) You also assume modal total loss factors even if they areconstant.

(5)Please illustrate mode shapes with natural frequencies andfrequency spectra of loss factors, input power and soundradiation power.

The report will be summarized in A4 size PDF with less than 10pages and sent to Prof. Iwatsuki via OCW-i by August 17, 2020.

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