東京工業大学工学院機械系 教授

イワツキ ノブユキ

岩附 信行

大学院講義

「静粛工学」

Tokyo Institute of TechnologyDept. of Mechanical EngineeringSchool of Engineering

Prof. Nobuyuki Iwatsuki

Lecture for Graduate Students

Silent Engineering

Lecture syllabus

Course title： Silent EngineeringAcademic major: Mechanical EngineeringOffered quarter: 2QDay/period: Tuesday 5-6Credits: 1-0-0Lecture room: ZOOMRegistration number: MECH.D532Lecturer: Prof. Nobuyuki Iwatsuki

Office: I1-305E-mail: iwatsuki.n.aa@m.titech.ac.jp

Course description and aims：The course offers the knowledge on the methods to quantitatively

estimate noise radiating from vibrating plates and to passivelyreduce the noise and includes the modal analysis and forcedvibration analysis of plates, the estimation of sound power radiatingfrom the plates based on the vibration analyses, the estimation ofsound power transmitting plates excited by sound, and the reductionof sound power with the structural optimization or dampingmaterials.

Because noise radiation from vibrating machinery strongly affectthe added value of the machinery, it is expected to reveal thepropagation mechanism from vibration to noise and to reduce noiseradiating from machinery. As the application of learning ofmechanical dynamics in Mechanical Engineering Course, studentswill understand the energy balance in vibrating plate and theestimation of frequency spectrum of sound radiation based on thevibration analysis and sound field analysis. Moreover, students willlearn the methods to reduce the sound radiation with the structuraloptimization or damping materials by taking account of costperformance.

Student learning outcomes：

By the end of this course, students will be able to:(1)Explain the outline of calculation process to analyze vibration

mode and forced vibration of plate(2)Explain the relation between the energy balance in vibrating

plate and the sound radiation power(3)Explain the outline of the methods to estimate frequency

spectrum of sound power radiating from vibrating plate(4)Explain the structural optimization to reduce noise radiation

(5)Explain the principle to reduce vibration with damping materials

Keywords：Vibration and noise, sound radiation power, modal analysis, forced vibration, parameters to estimate noise, structural optimization to reduce noise radiation, vibration damping

Class flow：Important issues are summarized at the end of lecture every week. Students are expected to understand what they learn by themselves.

Schedule：Class 1 Modal analysis and forced vibration analysis of plate(June 23) - Introduction of various methods to theoretically

analyze flexural vibration of plate -Class 2 Examples of analysis for various plates(June 30) - Fixing conditions of plates and the results of

modal analysis -Class 3 Formation of sound field due to vibrating source(July 7) - Point sound source and wave equation -

Class 4 Fundamental equations to estimate sound power (July 14) radiating from vibrating plate

- Energy balance in vibrating plate and parameters to estimate sound radiation power -

Class 5 Examples of estimation of sound power radiating (July 21) from vibrating plate

- Estimation of sound power radiating from rectangularplates and circular plates -

Class 6 Structural optimization to reduce sound radiation power(July 28) - Noise reduction by changing plate thickness or adding

ribs or hollows on plates -Class 7 Setting damping materials to reduce sound radiation (August 4) power

- Noise reduction with constraint and non-constraint dampers -

Textbook, reference book, course material：No textbook is required. Several handouts will be often distributedvia OCW-i. The following book is recommended as a reference book:(1)W. Weaver Jr., S. Timoshenko and D. H. Young,

Vibration Problems in Engineering (Fifth Edition),John Wiley and Sons (1990).

(2)K. Suzuki, K. Nishida, K. Maruyama and T. Watanabe, Vibrations and Acoustics for Mechanical Engineering, Science-Sha Co. Ltd. (2000) (in Japanese).

Assessment criteria and methods:Students' knowledge on the process to estimate the frequency spectrum of sound power radiating from vibrating plate based on accurate vibration analysis and sound radiation analysis and the methods to reduce sound power with structural optimization and damping optimization is assessed via submitted reports(100%) on several issues.

Related courses:Advanced Sound and Vibration Measurement, Experimental Modal Analysis for Structural Dynamics

Prerequisites:Students must have successfully completed 'Mechanical Vibration’or must have equivalent knowledge.

June 23, 2020

Silent Engineering(Lecture 1)

Modal analysis and forced vibration analysis of thin plate

-Introduction of various methods to theoretically analyze flexural vibration of plate -

Tokyo Institute of TechnologyDept. of Mechanical EngineeringSchool of Engineering

Prof. Nobuyuki Iwatsuki

1. Introduction of lecturerPlease visit my WEB-site:http://www.dynamics.mep.titech.ac.jp/index.php

Nobuyuki IwatsukiProfessor, Dr. Eng.

Affiliation:Department of Mechanical EngineeringSchool of EngineeringTokyo Institute of Technology

Biography:1978 Graduated from Kumamoto High School1981 Graduated from Dept. of Mech. Eng., Tokyo Tech.,

B. of Eng.1987 Graduated from Dept. of Mech. Eng., Graduate School of Sci.

and Eng., Tokyo Tech., Dr. of Eng.Research associate, Precision and Intelligent Laboratory,Tokyo Tech.

1995 Assoc. Prof. , Faculty of Eng., Tokyo Tech.1999 Visiting researcher at the Ohio State University and Stanford

University, USA2003 Prof., Graduate School of Sci. and Eng., Tokyo Tech.2016 Prof. and Dean, School of Eng., Tokyo Tech.2020 Prof., Dept. of Mechanical Eng., School of Eng., Tokyo Tech.

Research Themes:(A)Synthesis and Control of Robotic Mechanisms

Especially focused on ‘Mechanism Design and Motion Control of Hyper Redundant/UnderactuatedRobots’

(B)Silent EngineeringEspecially focused on ‘Estimation of Sound Power Radiating from Vibrating Structure and Structural Optimization to Reduce the Sound Power’

(C)Functional Material ActuatorsEspecially focused on ‘Development of Micro CiliaryActuators in Group’

Lorentz forceF

Magneticfield B

Z-coil

X-coil

Y-coil

GFRP

(1)Noise Estimation of MRI Device

MRI device

Introduction of Several Research Works

Frequency f Hz

Soun

d ra

diat

ion

pow

er W

rad

dBIn

put p

ower

Win

W0 400 800 1200 1600

10-15

10-10

10-5

100

0 400 800 1200 1600-20

20

60

100

CalculatedMeasured

Sound intensity measurement

0 500 1000 1500 2000-20

0

20

40

60

80MeasuredEstimated

Frequency [Hz]

Soun

d pr

essu

re le

vel [

dB]

200 coins(1 Jpn. Yen)are dropped from here.

Ensemble average

FEM Experimental

12th Mode

133rd ModeOrigin

deviated

(2)Estimation of Impact Noise of Coin Counting Machine

The calculated acceleration and sound pressure

Cross-section image by X-ray CT

(3)Estimation of Sound Radiation from Frog-type Guiro

Expanded

A steel drum which can generate musical scale

FEM model

Measured sound

Vibration mode

Calculated sound♪♪

♪ ♪

C

G

F

B

EAiming to create a new musical instrument

(4)Estimation of Musical Scale Generation of Steel Drum

2. Noise and Added Value of Machinery

Definition of Noise:Noise is a sound which is not desirable topersons and is the sound which blocks transfer of voices, music and so on orgives pain or injury to human ear.

－JIS Z 8106-1976

Classification of Noise:On propagation medium:○Air borne noise○Solid borne noise (Structure borne noise)

On sound source:○External noise source・Road traffic noise・Airplane noise・Machinery noise in factory

○Internal noise source・Construction equipment noise ・Work sound in factory・Human voice in office, noise from office machine・Crashing sound to floor of wall due to walking

or opening/closing of door

On time characteristics:○Steady noise

Noise of a continuous and almost constant level○Unsteady noise

・Time varying noiseNoise changing its level irregularly and continuously

・Intermittent noiseNoise generated intermittently with time interval

・Impact noiseNoise in which continuation time is very short

○Back ground noiseNoise other than the target of measurement

Noise and added value of machinery:Amount-of-money evaluation per 1dB noise in a judicial precedent

Reparations： 1,500JPY/month/person

65dB (Regulation value 45dB)

5 person family

Ex.: 1,500JPY×20dB×5 persons = 150,000JPY/month( 1,800,000 JPY / year )

Noise reduction expense increases in proportion to the square of a noise reduction rate!

90dB (-20dB) 85dB (-25dB)

110dB

5,000,000JPY

JPY000,000,5dB20dB25

JPY000,800,72

×

=

Noise reduction

Noise reduction

Example of machinery noise

Automobile:Road noise Increase of mass Increase of Engine noise Add damping materials fuel consumption

３. Machinery Noise

Excitation by the interaction of electromagnetic forces

Lorentz force F

Magnetic field B

Z coil

X coil

Y coil

GFRPMagnetic Resonant Imaging Device

Noise office machines

Coin counting machine

Cash-receipt-and-disbursement machine

Noises due to collision and conveyance of bills and coins

Bill sorting machine

Noise power radiating from vibrating plate or shellas a case of machinery

Noise radiationVibration propagation to the air

Exciting forceMechanical excitationAcoustic excitation

Plate/shell structure

Vibration propagationto other structure

↓Multi-DOFvibration system

“Flexural vibrationdue to natural mode ofvibration”

Vibration energy and radiating sound power

Input power Win＝FV

Sound (noise) radiation power Wrad

Energy of steady vibration E

Power to other structure Wext（negligible if stiff connection）

Internal dissipated power Wint（Dissipated as heat）

Velocity at exciting point

Calculated with acceleration of vibrating plane

Viscous damping（depends on velocity

“Accurately calculatedwith vibration response”

Wrad＝ηradωE＜Wint

Therefore, the forced vibration response of thin plate or shell should be analyzed with an adequate accuracy based on theoretical modal analysis of the plate or shell.

４. Forced Vibration Analysis of Thin Plate

４.1 Equation of vibration Assumptions:(1)Thickness, h , of plate is enough small than side length.(2)Cross-section before deformation is kept after deformation.

“Kirchhoff’s assumption”(3)Neutral plane is not extended.(4)External force, p(x,y), is distributed perpendicular to the

plate.(5)Shear deformation and rotational inertia are negligible.

“They cannot be ignored for thick plate.”Reissner-Mindlin Theory

Forces and moments applied on thin plate

Young’s modulus: EPoison’s ratio: νDensity: ρ p(x,y)

Distributed external force Bending moment

per unit width

Shearing force per unit width

Torsional moment per unit width

Force and moment balance:

02

2

=+−∂

∂+

∂∂

dtwdhp

yQ

xQ yx ρ

Force balance in z direction:

0=+∂

∂−

∂

∂y

yxy Qy

Mx

MMoment balance about x-axis:

0=−∂∂

+∂

∂x

xyx Qx

My

MMoment balance about y-axis:

Inertial force

(1)

(2)

(3)

By substituting Eqs.(2) and (3) into Eq.(1), we obtain

2

22

2

2

2

2

2twhp

yxM

yM

xM xyyx

∂∂

−=∂∂

∂−

∂

∂+

∂∂

ρ (４)

Shearing forces and moments per unit width:

dzQ

dzQh

h yzy

h

h xzx

∫∫

−

−

=

=2/

2/

2/

2/

τ

τShearing force:

zdzM

dzzMh

h yzy

h

h xzx

∫∫

−

−

=

=2/

2/

2/

2/

σ

σBending moment: zx

σx

xy

h

h yxyx

h

h xyxy

MzdzM

dzzM

==

=

∫∫

−

−

2/

2/

2/

2/

τ

τTorsional moment:

(5)

(6)

(7) Neutral plane

Displacement in plate:

wzwywzzv

xwzzu

=∂∂

−=

∂∂

−=

)(

)(

)( (8)

(9)

(10)

w

w(z)

u(z) zxw∂∂

z

Before deformation

Strains:

yxwz

xzv

yzuz

ywz

yzvz

xwz

xzuz

xy

y

x

∂∂∂

−=∂

∂+

∂∂

=

∂∂

−=∂

∂=

∂∂

−=∂

∂=

2

2

2

2

2

2)()()(

)()(

)()(

γ

ε

ε (11)

(12)

(13)

Displacement in plate:

wzwywzzv

xwzzu

=∂∂

−=

∂∂

−=

)(

)(

)( (8)

(9)

(10)

w

w(z)

u(z) zxw∂∂

z

Before deformation

Strains:

yxwz

xzv

yzuz

ywz

yzvz

xwz

xzuz

xy

y

x

∂∂∂

−=∂

∂+

∂∂

=

∂∂

−=∂

∂=

∂∂

−=∂

∂=

2

2

2

2

2

2)()()(

)()(

)()(

γ

ε

ε (11)

(12)

(13)

Relation between stresses and strains:

)()1(2

)(

)]()([1

)(

)]()([1

)(

2

2

zEz

zzEz

zzEz

xyxy

xyy

yxx

γν

τ

νεεν

σ

νεεν

σ

+=

+−

=

+−

= (14)

(15)

(16)

By substituting Eqs.(8)-(16) into Eqs.(5)-(7),we obtain

Moments written with bending deformation:

(17)

(18)

(19)yx

wvDM

xw

ywDM

yw

xwDM

xy

y

x

∂∂∂

−−=

∂∂

+∂∂

−=

∂∂

+∂∂

−=

2

2

2

2

2

2

2

2

2

)1(

ν

ν

where

By substituting Eqs.(17)-(19) into Eq.(3),we obtain

)1(121 2

32/

2/

22 νν −

=−

= ∫−EhdzzED

h

h　

Bending stiffness of plate

ptwh

yw

yxw

xwD =

∂∂

+∂∂

+∂∂

∂+

∂∂

2

2

4

4

22

4

4

4

)2( ρ

Equation of flexural vibration of thin plate

(20)

(21)

4.2 Free vibration analysisModal analysis

0)2( 2

2

4

4

22

4

4

4

=∂∂

+∂∂

+∂∂

∂+

∂∂

twh

yw

yxw

xwD ρ

Equation of free vibration of plate:

Let assume the solution with separation of variables as)sincos)(,(),,( tBtAyxWtyxw ωω +=

Mode shape Harmonic solution

(22)

(23)

By substituting Eq.(23) into Eq.(22),we obtain

0)2( 24

4

22

4

4

4

=−∂∂

+∂∂

∂+

∂∂ Wh

yW

yxW

xWD ωρ (24)

We have to obtain function W(x,y) which satisfiesEq.(24) and boundary condition.

Ex. A simply supported rectangular plate

x

y

a

b

O

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

Boundary conditions:

0)(,0;,0

0)(,0;,0

2

2

2

2

2

2

2

2

=∂∂

+∂∂

−===

=∂∂

+∂∂

−===

xw

ywDMwby

yw

xwDMwax

y

x

ν

ν

　

　

Resultantly we obtain

0,0;,0

0,0;,0

2

2

2

2

=∂∂

==

=∂∂

==

yWWby

xWWax

　

　 (25)

(26)

Let assume vibration shape W(x,y) as

byn

axmCyxW mn

ππ sinsin),( =

Then the boundary conditions can be satisfied as0),(),()0,(),0( ==== bxWyaWxWyW

Wb

nyWW

am

xW

2

22

2

2

2

22

2

2

, ππ−=

∂∂

−=∂∂

　Since

0),()0,(),(),0(2

2

2

2

2

2

2

2

=∂

∂=

∂∂

=∂

∂=

∂∂

ybxW

yxW

xyaW

xyW

(25)

(26)

(27)

We have to confirm that W(x,y) satisfies Eq.(24).

By substituting Eq.(25)into Eq.(24),we obtain

0)2( 24

44

22

422

4

44

=−++ WhWb

nWba

nmWa

mD ωρπππ

Therefore

22

2

2

24

4

44

22

422

4

442 )()2(

bn

am

hD

bn

banm

am

hD

+=++=ρππππ

ρω

)( 2

2

2

22

bn

am

hD

+=ρ

πω

Thus if angular frequency is as

,

Eq.(24) can be satisfied.

(28)

Natural angular frequency

Nodal lines

Mode of vibration of simply supported rectangular thin plate

Mode shape of simply supported rectangular thin plateCan be represented with nodal lines.

4.3 Forced vibration analysis

tyxptwh

yw

yxw

xwD ωρ cos),()2( 2

2

4

4

22

4

4

4

=∂∂

+∂∂

+∂∂

∂+

∂∂

Equation of forced vibration of plate:

(29)

Harmonic excitationNatural angular frequency : ωiMode shape （Eigenfunction): Wi(x,y)

have been calculated through free vibration analysis (modal analysis) where i means m,n

Let assume solution of forced vibration as linear combination of eigenfunction as

),()(),,( yxWtFtyxw ii

i∑= (30)

Time function Eigenfunction

By substituting Eq.(30)into Eq.(29),we obtain

tyxpyW

yxW

xWDFWFh

i

iiiii

ii ωρ cos),()2( 4

4

22

4

4

4

=∂∂

+∂∂

∂+

∂∂

+∑∑

Since Wi satisfies Eq.(24),

WhyW

yxW

xWD

yW

yxW

xWDWh

iiii

iiiii

24

4

22

4

4

4

4

4

22

4

4

42

)2(

0)2(

ωρ

ωρ

=∂∂

+∂∂

∂+

∂∂

∴

=∂∂

+∂∂

∂+

∂∂

+−

(31)

(32)

By substituting Eq.(32)into Eq.(31),we obtain

tyxpWFhWFhi

iiiii

i ωωρρ cos),(2 =+ ∑∑ (33)

Let rewrite Eq.(33) as matrix form as

[ ] [ ] tyxp

F

FF

WWWh

F

FF

WWWh

N

NN

N

N ωωωωρρ cos),(2

1

22

221

21

2

1

21 =

+

W SFF・・

tyxphh ωρρ cos),(=+∴ SFW (34)

By multiplying WT from left side and integrating the equation in whole Plate, we obtain

dstyxpdshdsh TTT ∫∫∫ =+SSSWSFWFWW ωρρ cos),(

( ) ( ) ( ) tdsyxpdshdsh TTT ωρρ cos),(∫∫∫ =+∴SSSWFSFWFWW

N×N matrix N×N matrix N×1 vector

(35)

F

F

+

∫∫

∫∫∫∫∫

∫∫

∫∫∫∫∫

S NNS N

SS

S NNSS

S NS N

SS

S NSS

dsWdsWW

dsWdsWW

dsWWdsWWdsW

h

dsWdsWW

dsWdsWW

dsWWdsWWdsW

h

221

21

22

2212

21

12

2122

21

21

21

2212

1212

1

ωω

ωω

ωωω

ρ

ρ

Let rewrite Eq.(35) as

t

dsyxpW

dsyxpW

dsyxpW

S N

S

S

ωcos

),(

),(

),(

2

1

=

∫

∫∫

(36)

00

00

Because of orthogonality of eigenfunction

( ) ( ) tdsyxpWdsWdiaghdsWdiaghS iS iiS i ωωρρ cos),(222

=⋅+⋅ ∫∫∫

FF

Let set as

∫∫

∫

=

==

=

S ii

iiS iii

S ii

dsyxpWQ

mdsWhk

dsWhm

),(

222

2

ωωρ

ρ Modal mass

Modal stiffness

Modal load

We can obtain

( ) ( ) tQkdiagmdiag iii ωcos

=+

FF

(37)

(38)

(39)

(40)

(41)

Therefore

）１～　　　 ＮitQFkFm iiiii ==+ (cosω (42)

Decoupled undamped forced vibration equation

Let adopt viscous damping as proportional damping as

( ) ( ) ( ) tQkdiagcdiagmdiag iiii ωcos

=++

FFF

where iiiiii mkmc ωζ2==

Modal damping ratio

(43)

(44)

Note that this is a important assumptionto make equation decoupled

ThereforetQFkFcFm iiiiiii ωcos=++ (45)

By normalizing the modal mass as unity as12 == ∫S ii dsWm

We obtain

)1(cos2 2 NitQFFF iiiiiii ～　　==++ ωωωζ

(46)

(47)

Decoupled damped forced vibration equation

Based on the solution of damped forced vibration system with 1 DOF,we can obtain

( )( ) ( )

)1(2tan

2

cos

221

2222

Ni

tQF

i

iii

iii

iii

～　　　　　　 =−

=

+−

−=

−

ωωωωζ

φ

ωωζωω

φω

where

(48)

(49)

Therefore the vibration displacement of thin plate can be represented as

( )( ) ( )

)1(2tan

2

cos),(

),()(),,(

221

2222

Ni

tyxWQ

yxWtFtyxw

i

iii

iiii

iii

iii

～　　　　　　

　　　　

=−

=

+−

−=

⋅=

−

∑

∑

ωωωωζ

φ

ωωζωω

φω

where

(50)

If the plate is excited by a force, P(t), at a certain point(xd,yd) , the distributed external force, p(x,y,t) can berepresented as

　

)()()(),,( dd yyxxtPtyxp −−= δδ

where δ denotes Dirac’s delta function

We can calculate vibration displacement, velocity and acceleration of thin plate

),()(),( ddS iii yxWtPdsyxpWQ ∫ ==

Therefore modal load, Qi, can be calculated as

5. Concluding remarksAs an introduction of silent engineering, the following issues are explained. (1)Noise radiating from machinery often determine

the added value of machinery.(2)It is important to accurately analyze the

vibration of thin plate or shell which covermachinery.

(3)Vibration equation of thin plate was derived.(4)Forced vibration of thin plate can be calculated

with the decoupled equation through modal analysis of the plate.